[Title]
102_̌vZ


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[Problem]
̎̌vZB

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5

[Level1]
̉@ƌ@
$(a-4b)-(2a-9b)$

$-a+5b$

$4(3a-b)-5(2a+b)$

$2a-9b$

$3x-3-4x-2$

$-x-5$

$x^2+5x+2x^2-x$

$3x^2+4x$

$-3(4x-2)$

$-12x+6$

$2x-5-6x+4$

$-4x-1$

$2x^2-5x-x^2+7x$

$x^2+2x$

$-2(3x-1)$

$-6x+2$

$4a-9-(a-5)$

$3a-4$

$a-2b+4a-6b$

$5a-8b$

$(6x-7y)-(4x+y)$

$2x-8y$

$3(3a-1)-(4a-7)$

$5a+4$

$4(3a-b)-(-3a+4b)$

$15a-8b$

$5a+9b-3(a+4b)$

$2a-3b$

$3(x+2y)+2(x-y)$

$5x+4y$

$2(6x-5y)-3(3x+7y)$

$3x-31y$

$3(7a+6)-4(5-8a)$

$53a-2$

$-5(x-2y)+3(x-3y)$

$-2x+y$

$3(x^2+2x-4)-2(3x-5)$

$3x^2-2$

$2(-a+5b-3)-(3a+7b-6)$

$-5a+3b$

$10\left( \displaystyle \frac{3}{5}a-\displaystyle \frac{1}{2}b \right)-2(a+3b)$

$4a-11b$

$6+3a+a^2-(7-a+a^2)$

$4a-1$

$7+5a+3a^2-(6-a+3a^2)$

$6a+1$

$(3x+2y)+(5x-6y)$

$8x-4y$

$(2x^2-4x+3)-(4x^2-7x-8)$

process
$2x^2-4x+3-4x^2+7x+8$

$-2x^2+3x+11$

$(3x+2y)-(5x-4y)$

$-2x+6y$

$-(7x-3y)-(x+8y)$

$-8x-5y$

$(4x^2-3)-(2x^2-6x-8)$

$2x^2+6x+5$

$-(5x^2+7x+9)+(8x^2-3x)$

$3x^2-10x-9$

$2x+\{ 4y-(3x-5y) \}$

$-x+9y$

$6x-\{ 9y-(4x-6y) \}$

process
$6x-\{9y-4x+6y\}$

$10x-15y$

$\left( \displaystyle \frac{1}{5} a- \displaystyle \frac{2}{3} b \right)+\left( \displaystyle \frac{3}{2} a- \displaystyle \frac{1}{6} b \right)$

process
$\displaystyle \frac{2}{10} a+\displaystyle \frac{15}{10} a-\displaystyle \frac{4}{6} b-\displaystyle \frac{1}{6} b$

$\displaystyle \frac{17}{10} a-\displaystyle \frac{5}{6} b$

$\left( -\displaystyle \frac{1}{4} a + b \right)- \left( \displaystyle \frac{1}{6} a -\displaystyle \frac{3}{7} b \right)$

process
$-\displaystyle \frac{1}{4} a+b-\displaystyle \frac{1}{6} a+\displaystyle \frac{3}{7} b$

$-\displaystyle \frac{5}{12} a+\displaystyle \frac{10}{7} b$

$4(5a+3b)+3(2a-4b)$

process
$20a+12b+6a-12b$

$26a$

$3(-2a-4b)-5(3b-4a)$

process
$-6a-12b-15b+20a$

$14a-27b$

$A=3x^2-4x, B=-x^2+5,C=2x-1$\\
̂Ƃ,\,$A-B+3C$\,vZȂB

$4x^2+2x-8$

$A=x^2+5x$C$B=2x^2-1$C$C=x^2+3x+4$̂ƂC$A+2B-C$vZB

process
\begin{eqnarray}
A &=& x^2+5x\\
2B &=& 4x^2\ \ \ \ \ \ \ -2\\
-C &=& -x^2-3x-4\\
A+2B-C &=& 4x^2+2x-6
\end{eqnarray}

$4x^2+2x-6$

$A=5x^3-6x^2+3x$C$B=3x$C$C=x^2-3x-2$̂ƂC$A-BC$vZB

process
$-BC=-3x(x^2-3x-2)\\
=-3x^3+9x^2+6x$C\\
$A=5x^3-6x^2+3x$\\
$A-BC=2x^3+3x^2+9x$

$2x^3+3x^2+9x$

$A=2x^2-x-2$C$B=x^2-2x$̂ƂC$-4A+B$vZB

process
$ -4A = -8x^2+4x+8$\\
$ \ B = x^2-2x$\\
$-4A+B=-7x^2+2x+8$

$-4A+B=-7x^2+2x+8$

$A=x-5,\,B=2x^2+x,\,C=-x^3+2$̂ƂC$AB-C$vZ\\
$\fbox{A}\,x^3-\fbox{C}\,x^2-\fbox{E}\,x-\fbox{G}$\\
ɂȂB

process
^$=x(x-5)(2x+1)+x^3-2\\
=x(2x^2-9x-5)+x^3-2$

AF3CCF9CEF5CGF2

$3a^3-2a^2+a-6-(a^3-2a^2+5)$

process
$3a^3-2a^2+a-6-a^3+2a^2-5=2a^3+a-11$

$2a^3+a-11$

$(3x^3-x^2-5x+7)+(8-4x-2x^2-x^3)$

$2x^3-3x^2-9x+15$

$2(4x^3-2x^2+x-1)+3(x^3-3x^2+4x+5)$

process
$8x^3-4x^2+2x-2+3x^3-9x^2+12x+15$

$11x^3-13x^2+14x+13$

$4(3x^2-4x+x^3-5)-2(-x+4x^2-7-2x^3)$

process
$4(x^3+3x^2-4x-5)-2(-2x^3+4x^2-x-7)\\
=4x^3+12x^2-16x-20+4x^3-8x^2+2x+14$

$8x^3+4x^2-14x-6$

$A=2x^2+3x-5$C$B=-4x^2+2x-8$̂ƂC$2A-B$vZB

process
\begin{eqnarray}
2A &=& 4x^2+6x-10\\
-B &=& 4x^2-2x+8\\
2A-B &=& 8x^2+4x-2
\end{eqnarray}

$8x^2+4x-2$

$A=2x^2+3x-5$C$B=-4x^2+2x-8$̂ƂC$A+B-3A$vZB

process
\begin{eqnarray}
-2A &=& -4x^2-6x+10\\
B &=& -4x^2+2x-8\\
-2A+B &=& -8x^2-4x+2
\end{eqnarray}

$-8x^2-4x+2$

$A=2x^2+3x-5$C$B=-4x^2+2x-8$̂ƂC\\
$3B-2A+2(A-B)$vZB

process
$3B-2A+2(A-B)=B$

$-4x^2+2x-8$

$7a-4b+5b-2a$

$5a+b$

$6a^2+a-5a^2-2a$

$a^2-a$

$(2x+5y)+(4x-y)$

$6x+4y$

$(2a-b)+(-5a-3b)$

$-3a-4b$

$7a-(3a-2b)$

$9a-3b$

$(4x-2y)-(x-3y)$

$3x+y$

$\left( 7ab-2b \right)+\left( -3ab-6b \right)$

$4ab-8b$

$\left( 9x-y \right)-\left( 4x-\displaystyle\frac{1}{5} y \right)$

process
$9x-y - 4x+\displaystyle\frac{1}{5} y$

$5x-\displaystyle\frac{4}{5} y$ 

$2x+y-\left( x-3y+2 \right)$

process
$2x+y-x+3y-2$

$x+4y-2$

$4a-\{ 8b+\left( 3a-2b \right) \}$

process
$4a-8b-3a+2b$

$a-6b$

$\left( 5a-3b \right)-\left( 2a+b \right)+\left( 3a+6b \right)$

process
$5a-3b-2a-b;3a+6b$

$6a+2b$

$4x-5y+2\left( x-3y \right)$

process
$4x-5y+2x-6y$

$6x-11y$

$3\left( x+2y \right)-2\left( 2x+5y \right)$

process
$3x+6y-4x-10y$

$-x-4y$

$5\left( 2x-3y \right)-3\left( 3x-2y \right)$

process
$10x-15y-9x+6y$

$x-9y$

$4\left( 2a-b \right)-3\left( 5a-2b \right)$

process
$8a-4b-15a+6b$

$-7a+2b$

$\displaystyle\frac{1}{2}a-\displaystyle\frac{2}{3}b-\left( \displaystyle\frac{2}{3}a-\displaystyle\frac{1}{4}b \right)$

process
$=\displaystyle\frac{1}{2}a-\displaystyle\frac{2}{3}a- \displaystyle\frac{2}{3}b+\displaystyle\frac{1}{4}b $\\
$=\displaystyle\frac{3}{6}a-\displaystyle\frac{4}{6}a- \displaystyle\frac{6}{4}b+\displaystyle\frac{1}{4}b $

$-\displaystyle\frac{1}{6}a-\displaystyle\frac{5}{4}b$

$3\left( x^2-3x-4 \right)-2\left( x^2+2x-5 \right) +\left( -x^2+8 \right)$

process
$3x^2-9x-12-2x^2-4x+10-x^2+8$

$-13x+6$

$8a\left( 2a+1 \right)-4a\left( 3-a \right)$

process
$16a^2+8a-12a+4a^2$

$20a^2-4a$

$4x\left( 3x-2y \right)-2x\left( 6x-5y \right)$

process
$12x^2-8xy-12x^2+10xy$

$2xy$

$A=2x+y+1$C$B=x+2y+2$C$C=x-2y-3$̂ƂC$A-2(B-C)$vZȂB

process
$2x+y+1-2(4y+5)=2x+y+1-8y-10$

$2x-7y-9$

$3x^2-5x+6$$A$$-x^2-2x+1$ƂȂB
̂ƂC$A$߂B

process
$3x^2-5x+6-A=-x^2-2x+1$\\
$A=3x^2-5x+6-(-x^2-2x+1)$\\
$=3x^2-5x+6+x^2+2x-1$\\
$=4x^2-3x+5$

$4x^2-3x+5$

$A=3x^2-x+2$C$B=x^2-3x+1$C\\
$C=-2x^2+5x-4$̂ƂC
$(B-A)+(A-C)$vZ
\fbox{\bf\, A \,}ɂȂB\\
\textcircled{\small 1}\,$2x^2-8x+5$~~~
\textcircled{\small 2}\,$2x^2-2x+5$\\
\textcircled{\small 3}\,$3x^2-8x+5$~~~
\textcircled{\small 4}\,$3x^2-2x+5$

\textcircled{\small 3}

$A$$-4x^2+5x+10$$2A$ƂȂB
̂Ƃ$A$߂B

process
$A+4x^2-5x-10=A$\\
$4x^2-5x-10=A$

$A=4x^2-5x-10$

$A=3x-1$C$B=5x^2+3x-1$C$C=x^2+6x-1$̂ƂC
$2A+B-2C$vZ
\fbox{\bf\, A \,}ɂȂB\\
\textcircled{\small 1}\,$3x^2-3x-14$~~~
\textcircled{\small 2}\,$3x^2-3x+9$\\
\textcircled{\small 3}\,$7x^2-3x-14$~~~
\textcircled{\small 4}\,$7x^2-3x+9$

\textcircled{\small 2}

$A$$x^2+2x-5$$2x^2-4x-1$ɂȂB
̂ƂC
$A=$\fbox{\bf\, A \,}$x^2-$\fbox{\bf\, C \,}$x-$\fbox{\bf\, E \,}łB

process
$A-(x^2+2x-5)=2x^2-4x-1$\\
$A=2x^2-4x-1+x^2+2x-5=3x^2-2x-6$

$3x^2-2x-6$





[Level2]
PƑ̏@E@
$6ab^2 \div \displaystyle \frac{2}{3}ab$

$9b$

$2x^2 \times 6xy^2 \div (-4xy)$

$-3x^2y$

$(8xy+4x) \div 4x$

$2y+1$

$2x^3 \times (-4xy)$

$-8x^4y$

$12xy \div 3x$

$4y$

$x^3 \times x^2$

$x^5$

$4x^3 \times (-2x^2)$

$-8x^5$

$x^6 \div x^2$

$x^4$

$18x^6 \div 6x^3$

$3x^3$

$(x^2)^4$

$x^8$

$(-5x^3)^2$

$25x^6$

$12x^3 \div 3x \times 4x$

$16x^3$

$(9xy+3x) \div 3x$

$3y+1$

$2x \times (-5xy^2)$

$-10x^2y^2$

$15xy \div 3y$

$5x$

$x^6 \times x^2$

$x^8$

$4x^3 \times (-2x^2)$

$-8x^5$

$x^8 \div x^2$

$x^6$

$24x^6 \div 6x^2$

$4x^4$

$(x^4)^2$

$x^8$

$(-3x^3)^2$

$9x^6$

$15x^3 \div 3x \times 5x^2$

$25x^4$

$5x^2 \times 8xy^3$

$40x^3y^3$

$(-3a)^2 \times 6ab$

$54a^3b$

$3a^2b \times \displaystyle \frac{1}{9} a$

$\displaystyle \frac{1}{3} a^3 b$

$\displaystyle \frac{4}{3}x^2y \times \displaystyle \frac{9}{2} y$

$6x^2y^2$

$-18a^2b^3 \div 3ab$

$-6ab^2$

$27a^2b \div (-9ab)$

$-3a$

$(-2a)^3 \div4a^2$

$-2a$

$(-6xy^2) \div \displaystyle \frac{1}{3}xy$

$-18y$

$x^2 \times 3y \div 6xy$

$\displaystyle \frac{1}{2} x$

$2xy^2 \div (-4y) \times (-6x)$

$3x^2y$

$15a^2b \div(-9ab^2) \times 3ab$

$-5a^2$

$6xy \times (-3x) \div (3x)^2$

$-2y$

$32x^3y^4 \div 8xy^2 \times (xy)^2$

$4x^4y^4$

$4xy^3 \times \left( \displaystyle \frac{3x}{y} \right)^2 \div 2x^2$

$18xy$

$\displaystyle \frac{b}{2} \times (-ab)^2 \div \left( -\displaystyle \frac{1}{2} b \right)^2$

$2a^2b$

$10a^2b \div (-3ab) \times \left( \displaystyle \frac{3}{5} b \right)^2$

$-\displaystyle \frac{6}{5} ab^2$

$15x^2 \div (-3x)$

$-5x$

$18x^2y \div \displaystyle \frac{3}{4} x$

$24xy$

$5a \div 9ab \times 3a^2b$

$\displaystyle \frac{5}{3} a^2$

$12xy^2 \times xy \div (-6x^2y)$

$-2y^2$

$10a \times 6a^2 \div (2a)^2$

$15a$

$15a^3 \times 6a \div (-3a)^2$

$10a^2$

$(-3ab)^3 \div 9a^2b \div (-ab)$

$3b$

$32a^2b^2 \div (-8ab) \div(-4b)$

$a$

$\displaystyle \frac{1}{3} x \times \left( -\displaystyle \frac{3}{4} x^2y \right)$

$-\displaystyle \frac{x^3y}{4}$

$14xy \div 7y$

$2x$

$8x^2y \div \displaystyle \frac{4x}{y}$

process
$8x^2y \times \displaystyle \frac{y}{4x}$

$2xy^2$

$2a(3x-5y)$

$6ax-10ay$

$(4x-6y)\times (-3b)$

$-12bx+18by$

$(9a^2-6a)\div 3a $

$3a-2$

$(24b^3-18b^2-15b) \div (-3b)$

$-8b^2+6b+5$

$3(2a-5b)$

$6a-15b$

$(-3x^2-5x+7) \times (-4)$

$-12x^2+20x-28$

$(12x^2-9x) \div(-3x)$

$-4x+3$

$(-18a^2+15ab-3b^2) \div \displaystyle \frac{3}{2}$

process
$(-18a^2+15ab-3b^2) \times \displaystyle \frac{2}{3}$

$-12a^2+10ab-2b^2$

$(-3a) \times(-2a)$

$6a^2$

$-4a^2 \times 3a$

$-12a^3$

$\displaystyle \frac{1}{2} x^2y \times xy$

$\displaystyle \frac{1}{2} x^3y^2$

$\displaystyle \frac{5}{6} a \times (-3b^2)$

$-\displaystyle \frac{5}{2} ab^2$

$(-8xy^2) \div 4xy$

$-2y$

$12a^2 \div(-6ab)$

$-\displaystyle \frac{2a}{b}$

$(-6x^2) \div 3x \times (-9x)$

$18x^2$

$8a^2b \div (-a)^2 \times 2b$

$16b^2$

$5x(-2x^2y-3xy^2)$

$-10x^3y-15x^2y^2$

$(3x^2y-7xy) \times (-4xy)$

$-12x^3y^2+28x^2y^2$

$-6xy \left( \displaystyle \frac{2}{3} y^2 -\displaystyle \frac{3}{4} x^2 \right)$

$-4xy^3+\displaystyle \frac{9}{2} x^3y$

$(-4a^2+12b) \times \left( -\displaystyle \frac{1}{4} c \right)$

$a^2c-3bc$

$(10ax-14ay) \div 2a$

$5x-7y$

$(18a^3b^2+21ab^2) \div 3ab $

$6a^2b+7b$

$(-12x^2y+30x^3y) \div \displaystyle \frac{1}{2} x^2y$

process
$(-12x^2y+30x^3y) \times \displaystyle \frac{2}{x^2y}$

$-24+60x$

$\left( \displaystyle \frac{2}{3} x^3y - \displaystyle \frac{1}{6}xy^2 \right) \div \displaystyle \frac{1}{6} xy$

process
$\left( \displaystyle \frac{2}{3} x^3y - \displaystyle \frac{1}{6}xy^2 \right) \times \displaystyle \frac{6}{xy}$

$4x^2-y$

$(2x^5 \times 3x^3)^2$vZB

process
$(6x^8)^2=36x^{16}$

$36x^{16}$

$(4xy)(2xy)^3 \div 8x^2y^3$

process
$\displaystyle \frac{4xy \times 8x^3y^3}{8x^2y^3}=4x^2y$

$4x^2y$

$3xy^2\times 5x^2y^3$

$15x^3y^5$

$2x^3y\times (-4x^2y^3)$

$-8x^5y^4$

$(-2x^2y)^3$

$-8x^6y^3$

$(-3x)\times (-xy)^2$

process
$(-3x)\times (x^2y^2)$

$-3x^3y^2$

$xy^2(x+2xy+3)$

$x^2y^2+2x^2y^3+3xy^2$

$(4a-2b+3c)\times 2ab^2$

$8a^2b^2-4ab^3+6ab^2c$

$3a^2b^3 \times 5ab^4$

$15a^3b^7$

$(-2x^2y)^2\times (-3x^3y^2)^3$

process
$4x^4y^2\times (-27x^9y^6)$

$-108x^{13}y^8$

$(8a^3b^4)\div (-2a^2b^2)$

$-4ab^2$

$(-4ax)^3\div (-4ax^2y)$

process
$\displaystyle\frac{-64a^3x^3}{-4ax^2y}$

$\displaystyle\frac{16a^2x}{y}$

$(-3xy^2)^2\div (9x^3y^5)\times (-2xy^2)^3$

process
$\displaystyle\frac{9x^2y^4\times (-8x^3y^6)}{9x^3y^5}$

$-8x^2y^5$

$x^4y^3\times (x^2yz^3)^2$

process
$x^4y^3\times x^4y^2z^6$

$x^8y^5z^6$

$\left( 2x-6y-4 \right) \times \left( -\displaystyle\frac{1}{2} \right)$

$-x+3y+2$

$\left( 4a+18b \right) \div 2 $

$2a+9b$

$\left( 12x-18y+6z \right) \div \left(-6 \right)$

$-2x+3y-z$

$\left( 6x^2-15xy+9y^2 \right) \div \displaystyle\frac{3}{5}$

$10x^2-25xy+15y^2$

$-4 \left(-2a+3b \right)$

$8a-12b$

$\displaystyle\frac{5}{9} a \times \left( -27b \right)$

$-15ab$

$\left( -\displaystyle\frac{3}{10} a \right) \times \left( -\displaystyle\frac{5}{6} b \right)$

$\displaystyle\frac{1}{4} ab$

$12ab \div \left( -\displaystyle\frac{6}{7} b \right)$

process
$12ab \times \left( -\displaystyle\frac{7}{6b} \right)$

$-14a$

$-3a\times \left( -2a \right)^2$

process
$=-3a\times 4a^2$

$-12a^3$

$21ab^3\div \left( -3b^2 \right)$

$-7ab$

$\left( \displaystyle\frac{1}{2} xy\right)^2\times \displaystyle\frac{2}{3} x^2y$

process
$\displaystyle\frac{x^2y^2}{4} \times \displaystyle\frac{2}{3} x^2y$

$\displaystyle\frac{x^4y^3}{6}$

$6x^3y \div 2xy \times 3x$

process
$\displaystyle\frac{6x^3y}{2xy} \times \displaystyle\frac{3x}{1}$

$9x^3$

$9x^2y^2\div \left( -3x \right) \div xy$

process
$\displaystyle\frac{9x^2y^2}{-3x} \times \displaystyle\frac{1}{xy}$

$-3y$

$4a^2b\times \left( 2a^2 \right)^3$

process
$=4a^2b \times 8a^6$

$32a^8b$

$-\displaystyle\frac{3}{4} x\times \left( -3y \right)^2\div \displaystyle\frac{9}{8}x$

process
$=-\displaystyle\frac{3x}{4} \times \displaystyle\frac{9y^2}{1} \times\displaystyle\frac{8}{9x}$

$-6y^2$

$\left( 9x^2-3x \right) \div 3x$

$3x-1$

$\left( 2xy-4y \right) \div \displaystyle\frac{2}{3}y$

process
$=\left( 2xy-4y \right) \times \displaystyle\frac{3}{2y}$\\
$=2xy \times \displaystyle\frac{3}{2y} -4y\times \displaystyle\frac{3}{2y}$

$3x-6$

$5x\times 3xy+4xy\left( 6-x \right)$

process
$15x^2y+24xy-4x^2y$

$11x^2y+24xy$









[Level3]
̌`̎̌vZ
$\displaystyle \frac{x}{2}-\displaystyle \frac{x}{3}$

$\displaystyle \frac{x}{6}$

$\displaystyle \frac{x+3}{2}+\displaystyle \frac{x+2}{5}$

$\displaystyle \frac{7x+19}{10}$

$\displaystyle \frac{x+2}{3}-\displaystyle \frac{x-1}{6}$

process
$\displaystyle \frac{2(x+2)-(x-1)}{6}$

$\displaystyle \frac{x+5}{6}$

$\displaystyle \frac{x}{4}-\displaystyle \frac{x}{6}$

$\displaystyle \frac{x}{12}$

$\displaystyle \frac{x+4}{3}+\displaystyle \frac{x+1}{4}$

$\displaystyle \frac{7x+19}{12}$

$\displaystyle \frac{x+3}{2}-\displaystyle \frac{x-1}{4}$

$\displaystyle \frac{x+7}{4}$

$\displaystyle \frac{3x-5y}{2}-\displaystyle \frac{4x-7y}{3}$

$\displaystyle \frac{x-y}{6}$

$\displaystyle \frac{1}{8} (7x-4)-\displaystyle \frac{1}{2}(x-1)$

process
$\displaystyle \frac{1}{8} (7x-4)+\displaystyle \frac{1}{8}(-4x+4)$

$\displaystyle \frac{3}{8} x$

$\displaystyle \frac{1}{2} (x-4)-\displaystyle \frac{2}{5}(x-5)$

process
$\displaystyle \frac{1}{10} (5x-20)+\displaystyle \frac{1}{10}(-4x+20)$

$\displaystyle \frac{1}{10} x$

$\displaystyle \frac{x}{2}-\displaystyle \frac{x-2y}{3}$

process
$\displaystyle \frac{3x}{6}+\displaystyle \frac{-2x+4y}{6}$

$\displaystyle \frac{x+4y}{6}$

$\displaystyle \frac{x-2y}{6}+\displaystyle \frac{x+y}{8}$

process
$\displaystyle \frac{4x-8y}{24}+\displaystyle \frac{3x+3y}{24}$

$\displaystyle \frac{7x-5y}{24}$

$\displaystyle \frac{3a+b}{4}-\displaystyle \frac{a-b}{6}$

process
$\displaystyle \frac{9a+3b}{12}+\displaystyle \frac{-2a+2b}{12}$

$\displaystyle \frac{7a+5b}{12}$

$\displaystyle \frac{x-2y}{3}-\displaystyle \frac{6x-y}{5}+x$

process
$\displaystyle \frac{5x-10y}{15}+\displaystyle \frac{-18x+3y}{15}+x
=\displaystyle \frac{-13x-7y}{15}+\displaystyle \frac{15x}{15}$

$\displaystyle \frac{2x-7y}{15} $

$\displaystyle \frac{5x-4}{6}-\displaystyle \frac{x+1}{3}-\displaystyle \frac{2x-5}{7}$

process
$\displaystyle \frac{5x-4}{6}+\displaystyle \frac{-2x-2}{6}+\displaystyle \frac{-2x+5}{7}
=\displaystyle \frac{x-2}{2}+\displaystyle \frac{-2x+5}{7}$

$\displaystyle \frac{3x-4}{14} $

$x-\displaystyle \frac{x-2y}{3}-\displaystyle \frac{2x-3y}{4}$

process
$x-\displaystyle \frac{-x+2y}{3}+\displaystyle \frac{-2x+3y}{4}\\
=x+\displaystyle \frac{-4x+8y}{12}+\displaystyle \frac{-6x+9y}{12}\\
=\displaystyle \frac{12x}{12}+\displaystyle \frac{-10x+17y}{12}$

$\displaystyle \frac{2x+17y}{12}$

$\displaystyle \frac{3a-2}{2}-\displaystyle \frac{5a-6}{6}$

process
$\displaystyle \frac{9a-6}{6}+\displaystyle \frac{-5a+6}{6}$

$\displaystyle \frac{2a}{3}$

$\displaystyle \frac{x-2y}{3}-\displaystyle \frac{x-3y}{4}$

process
$\displaystyle \frac{4x-8y}{12}+\displaystyle \frac{-3x+9y}{12}$

$\displaystyle \frac{x+y}{12}$

$12 \times \left( \displaystyle \frac{a-1}{3}-\displaystyle \frac{a-2}{6} \right) \div \displaystyle \frac{2}{5} a$

process
$12 \times \left( \displaystyle \frac{2a-2}{6}+\displaystyle \frac{-a+2}{6} \right) \times \displaystyle \frac{5}{2a}
=12 \times \displaystyle \frac{a}{6} \times \displaystyle \frac{5}{2a}$

$5$

$45 \times \left( \displaystyle \frac{2a-1}{5}-\displaystyle \frac{3a-2}{10}\right) \div \displaystyle \frac{3}{2}a$

process
$45 \times \left( \displaystyle \frac{4a-2}{10}+\displaystyle \frac{-3a+2}{10} \right) \times \displaystyle \frac{2}{3a} 
=45 \times \displaystyle \frac{a}{10} \times \displaystyle \frac{2}{3a}$

$3$

$3a+\displaystyle \frac{5a-3}{2}$

process
$\displaystyle \frac{6a}{2}+\displaystyle \frac{5a-3}{2}$

$\displaystyle \frac{11a-3}{2}$

$\displaystyle \frac{a+5b}{3}+\displaystyle \frac{5a-7b}{4}$

process
$\displaystyle \frac{4a+20b}{12}+\displaystyle \frac{15a-21b}{12}$

$\displaystyle \frac{19a-b}{12}$

$\displaystyle \frac{x-3y}{3}+\displaystyle \frac{2x+y}{4}$

process
$\displaystyle \frac{4x-12y}{12}+\displaystyle \frac{6x+3y}{12}$

$\displaystyle \frac{10x-9y}{12}=\displaystyle \frac{5}{6} x-\displaystyle \frac{3}{4} y$

$\displaystyle \frac{3a-4b}{5}-\displaystyle \frac{a-b}{6}$

process
$\displaystyle \frac{3a-4b}{5}+\displaystyle \frac{-a+b}{6}=\displaystyle \frac{18a-24b-5a+5b}{30}$

$\displaystyle \frac{13a-19b}{30}=\displaystyle \frac{13}{30} a-\displaystyle \frac{19}{30} b$

$2a+\displaystyle \frac{3a-8}{6}$

process
$\displaystyle \frac{12a}{6}+\displaystyle \frac{3a-8}{6}$

$\displaystyle \frac{15a-8}{6}=\displaystyle \frac{5}{2} a-\displaystyle \frac{4}{3}$

$-\displaystyle \frac{3x-2y}{2}+\displaystyle \frac{5x-3y}{4}$

process
$\displaystyle \frac{-6x+4y}{4}+\displaystyle \frac{5x-3y}{4}$

$\displaystyle \frac{-x+y}{4}=-\displaystyle \frac{x-y}{4}$

$\displaystyle \frac{2x-3y}{4}-\displaystyle \frac{x-2y}{3}$

process
$\displaystyle \frac{3(2x-3y)}{12}+\displaystyle \frac{-4(x-2y)}{12}\\
=\displaystyle \frac{6x-9y-4x+8y}{12}=\displaystyle \frac{2x-y}{12}$

$\displaystyle \frac{2x-y}{12}$

̎ȒPɂB\\
$\displaystyle \frac{1}{1-\displaystyle \frac{1}{1+\displaystyle \frac{1}{x}}}$

process
^$=\displaystyle \frac{1}{1-\displaystyle \frac{x}{x+1}}=\displaystyle \frac{x+1}{x+1-x}=x+1$

$x+1$

$\displaystyle\frac{1}{2} x-y+\displaystyle\frac{2}{3} x + \displaystyle\frac{3}{4} y$

process
$\displaystyle\frac{3}{6} x +\displaystyle\frac{4}{6} x + \displaystyle\frac{3}{4} y-\displaystyle\frac{4}{4}y$

$\displaystyle\frac{7}{6} x-\displaystyle\frac{1}{4} y$

$a-2b-\displaystyle\frac{2a+b}{3}$

process
$\displaystyle\frac{3a-6b}{3}+\displaystyle\frac{-2a-b}{3}$

$\displaystyle\frac{a-7b}{3}$

$\displaystyle\frac{3a-b}{5}-\displaystyle\frac{a-3b}{3}$

process
$\displaystyle\frac{9a-3b}{15}+\displaystyle\frac{-5a+15b}{15}$

$\displaystyle\frac{4a+12b}{15}$

$2x-\displaystyle\frac{x-y}{8}-\displaystyle\frac{2y-x}{2}$

process
$=2x+\displaystyle\frac{-x+y}{8}+\displaystyle\frac{8y-4x}{8}$\\
$=\displaystyle\frac{16x}{8}+\displaystyle\frac{-5x+9y}{8}$

$\displaystyle\frac{11x+9y}{8}$



[Level4]
WJibj
̎WJȂB\\
$(x-2y)^2$

$x^2-4xy+4y^2$

̎WJȂB\\
$(x-3)^2$

$x^2-6x+9$

̎WJȂB\\
$(a+2)(a-2)$

$a^2-4$

̎WJȂB\\
$(x+3)(x-4)$

$x^2-x-12$

̎WJȂB\\
$(x-4)^2$

$x^2-8x+16$

̎WJȂB\\
$(a+3)(a-3)$

$a^2-9$

̎WJȂB\\
$(x-2)(x+5)$

$x^2+3x-10$

̎WJȂB\\
$(x+1)(x+3)-(x+2)^2$

process
$x^2+4x+3-(x^2+4x+4)=x^2+4x+3-x^2-4x-4$

$-1$

̎WJȂB\\
$(x+1)(x-1)-(x-1)^2$

process
$x^2-1-(x^2-2x+1)=x^2-1-x^2+2x-1$

$2x-2$

̎WJȂB\\
$(2x-4)(3x+7)$

$6x^2+2x-28$

̎WJȂB\\
$(4a+5)(5a-2)$

$20a^2+17a-10$

̎WJȂB\\
$\left( \displaystyle \frac{1}{2}a-\displaystyle \frac{1}{4}b \right) \left( \displaystyle \frac{2}{3}a-\displaystyle \frac{1}{6}b \right)$

process
$\displaystyle \frac{1}{3}a^2-\displaystyle \frac{1}{12}ab -\displaystyle \frac{1}{6} ab +\displaystyle \frac{1}{24} b^2$

$\displaystyle \frac{1}{3}a^2-\displaystyle \frac{1}{4}ab +\displaystyle \frac{1}{24} b^2$

̎WJȂB\\
$(x+3)(x+4)$

$x^2+7x+12$

̎WJȂB\\
$(x-2)(x+6)$

$x^2+4x-12$

̎WJȂB\\
$(2x-4)(3x+5)$

$6x^2-2x-20$

̎WJȂB\\
$(-4x+5)(6x-8)$

$-24x^2+62x-40$

̎WJȂB\\
$(2a+b)(a+3b)-(a-3b)(2a-4b)$

process
$2a^2+7ab+3b^2-(2a^2-10ab+12b^2)=2a^2+7ab+3b^2-2a^2+10ab-12b^2$

$17ab-9b^2$

̎WJȂB\\
$(x^2+3x-2)(4+x^2)$

process
$4x^2+12x-8+x^4+3x^3-2x^2$

$x^4+3x^3+2x^2+12x-8$

̎WJȂB\\
$(2a+4)^2$

$4a^2+16a+16$

̎WJȂB\\
$(3y-5)^2$

$9y^2-30y+25$

̎WJȂB\\
$(3a+2b)^2$

$9a^2+12ab+4b^2$

̎WJȂB\\
$(4x-6y)^2$

$16x^2-48xy+36y^2$

̎WJȂB\\
$(x+5)(x-5)$

$x^2-25$

̎WJȂB\\
$(9x+2y)(9x-2y)$

$81x^2-4y^2$

̎WJȂB\\
$(4x+2)(4x-7)$

$16x^2-20x-14$

̎WJȂB\\
$(x+3)^2-(x+6)(x-1)$

process
$x^2+6x+9-(x^2+5x-6)\\
=x^2+6x+9-x^2-5x+6$

$x+15$

̎WJȂB\\
$(x-1)(x^2+2x+1)$

process
$x^3+2x^2+x-x^2-2x-1$

$x^3+x^2-x-1$

̎WJȂB\\
$(3x-4)(-2x^2-x+6)$

process
$-6x^3-3x^2+18x+8x^2+4x-24$

$-6x^3+5x^2+22x-24$

̎WJȂB\\
$(2x+y)^2$

$4x^2+4xy+y^2$

̎WJȂB\\
$(3x+y)(3x-y)$

$9x^2-y^2$

̎WJȂB\\
$(x+7)(x+4)$

$x^2+11x+28$

̎WJȂB\\
$(x+8)(x-3)$

$x^2+5x-24$

̎WJȂB\\
$(2x-5)(3x+6)$

process
$6x^2+12x-15x-30$

$6x^2-3x-30$

̎WJȂB\\
$(5x+7)(4x-2)$

process
$20x^2-10x+28x-14$

$20x^2+18x-14$

̎WJȂB\\
$(x+3y)(x-4y)$

process
$x^2-4xy+3xy-12y^2$

$x^2-xy-12y^2$

̎WJȂB\\
$(3x+5y)(4x-6y)$

process
$12x^2-18xy+20xy-30y^2$

$12x^2+2xy-30y^2$

̎WJȂB\\
$(2a+3b)(3a+2b)$

$6a^2+13ab+6b^2$






[Level5]
WJipj
̎ȒPɂȂB\\
$(\sqrt{3}+\sqrt{2}+1)(\sqrt{3}-\sqrt{2}+1)$

process
$(\sqrt{3}+1+\sqrt{2})(\sqrt{3}+1-\sqrt{2})=(\sqrt{3}+1)^2-(\sqrt{2})^2$

$2+2\sqrt{3}$

̎WJȂB\\
$(a+b-1)(a+b+1)$

process
$a+b=x$Ƃ\\
$(x-1)(x+1)=x^2-1=(a+b)^2-1$

$a^2+2ab+b^2-1$

̎WJȂB\\
$(x+y+3)(x+y-2)$

process
$x+y=a$Ƃ\\
$(a+3)(a-2)=a^2+a-6=(x+y)^2+(x+y)-6$

$x^2+2xy+y^2+x+y-6$

̎WJȂB\\
$(x+y+2)(x+y-2)$

process
$x+y=a$Ƃ\\
$(a+2)(a-2)=a^2-4=(x+y)^2-4$

$x^2+2xy+y^2-4$

̎WJȂB\\
$(x+y+1)(x+y-1)$

process
$x+y=a$Ƃ\\
$(a+1)(a-1)=a^2-1=(x+y)^2-1$

$x^2+y^2+2xy-1$

̎WJȂB\\
$(a+b-c)^2$

process
$a+b=x$Ƃ\\
$(x-c)^2=x^2+c^2-2cx=(a+b)^2+c^2-2c(a+b)$

$a^2+b^2+c^2+2ab-2bc-2ca$

̎WJȂB\\
$(a^2+2a-3)(a-1)$

$a^3+a^2-5a+3$

̎WJȂB\\
$(x+4)(3x^2-2x+1)$

$3x^3+10x^2-7x+4$

̎WJȂB\\
$(x^3+x^2+x+1)(x-1)$

$x^4-1$

̎WJȂB\\
$(2x+3)^2-2(x+2)(2x-3)$

10x+21

̎WJȂB\\
$(2x+y)(x+2y)-2(x-y)^2$

$9xy$

̎WJȂB\\
$(2a+b)^2-(a+2b)(2a-b)$

$2a^2+ab+3b^2$

̎WJȂB\\
$(x-y+1)(x-y+2)$

$x^2-2xy+y^2+3x-3y+2$

̎WJȂB\\
$(x+y+z)^2-(x+y-z)^2$

$4xz+4yz$

̎WJȂB\\
$(a+b-c)(a+b+c)$

process
$(a+b)^2-c^2=a^2+b^2+2ab-c^2$

$a^2+b^2-c^2+2ab$

̎WJȂB\\
$(a+b+4)(a+b-6)$

$a^2+2ab+b^2-2a-2b-24$

̎WJȂB\\
$(x+2y+3)(x+5+2y)$

$x^2+4xy+4y^2+8x+16y+15$

̎WJȂB\\
$(x^2-xy+y^2)(x^2+xy+y^2)$

$x^4+y^4+x^2y^2$

̎WJȂB\\
$(x-3)(x-2)(x+2)(x+3)$

process
$(x^2-9)(x^2-4)$

$x^4-13x^2+36$

̎WJȂB\\
$(x^4+1)(x^2+1)(x+1)(x-1)$

process
$(x^4+1)(x^2+1)(x^2-1)$

$x^8-1$

̎WJȂB\\
$(a-b-c)^2$

process
ij1j\\
$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx$C\\
$(a-b-c)^2\\
=a^2+(-b)^2+(-c)^2\\
+2a(-b)+2(-b)(-c)+2(-c)a\\
=a^2+b^2+c^2-2ab+2bc-2ca$\\
ij2j\\
$(a-b-c)^2\\
=\{ a-(b+c)\}^2=a^2+(b+c)^2-2a(b+c)\\
=a^2+b^2+c^2+2bc-2ab-2ca$

$a^2+b^2+c^2-2ab+2bc-2ca$

̎WJȂB\\
$a-\{ 3b-2\left( 3a-2b \right) \}$

process
$a-\{ 3b-6a+4b \}=a-\{ -6a+7b \}=a+6a-7b$

$7a-7b$

̎WJȂB\\
$(a-2b+c)(a-2b-c)$

process
$(a-2b)^2-c^2=a^2-4ab+4b^2-c^2$

$a^2+4b^2-c^2-4ab$

̎WJȂB\\
$(a+b+c)(a-b-c)$

process
$a^2-(b+c)^2=a^2-(b^2+c^2+2bc)$

$a^2-b^2-c^2-2bc$

̎WJȂB\\
$(a+b-c)(a-b+c)$

process
$\{ a+(b-c) \}\{ a-(b-c) \}=a^2-(b-c)^2\\
=a^2-(b^2+c^2-2bc)=a^2-b^2-c^2+2bc$

$a^2-b^2-c^2+2bc$

̎WJȂB\\
$(a-b)^2(a+b)^2$

process
$\{(a-b)(a+b)\}^2=(a^2-b^2)^2\\
=a^4-2a^2b^2+b^4$

$a^4-2a^2b^2+b^4$

̎WJƁCǂ̎ɂȂ邩B
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
$(x-y+1)(x+y+1)$\\
\textcircled{\small 1}\,$x^2-y^2-1$\\
\textcircled{\small 2}\,$x^2-y^2-2y-1$\\
\textcircled{\small 3}\,$x^2-y^2+2y-1$\\
\textcircled{\small 4}\,$x^2-y^2+2x+1$\\

process
$\{ (x+1)-y \}\{ (x+1)+y \}\\
=(x+1)^2-y^2=x^2+2x+1-y^2$

\textcircled{\small 4}

̎WJƁCǂ̎ɂȂ邩B
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
$(x+y-1)(x-y+1)$\\
\textcircled{\small 1}\,$x^2-y^2-1$\\
\textcircled{\small 2}\,$x^2-y^2-2y-1$\\
\textcircled{\small 3}\,$x^2-y^2+2y-1$\\
\textcircled{\small 4}\,$x^2-y^2+2x+1$\\

process
$\{ x+(y-1) \}\{ x-(y-1) \}\\
=x^2-(y-1)^2=x^2-(y^2-2y+1)$

\textcircled{\small 3}

̎WJȂB\\
$(a+2b+c)(a-2b-c)$

process
$(a+2b+c)\{ a-(2b+c)\}=a^2-(2b+c)^2\\
=a^2-(4b^2+4bc+c^2)$

$a^2-4b^2-4bc-c^2$

̎WJȂB\\
$\left( x-\displaystyle\frac{1}{3} \right)\left( x+\displaystyle\frac{1}{5} \right)$

process
$x^2+\left( \displaystyle\frac{1}{5}-\displaystyle\frac{1}{3} \right)x-\displaystyle\frac{1}{15}$

$x^2-\displaystyle\frac{2}{15}x-\displaystyle\frac{1}{15}$

̎WJȂB\\
$\left( x+\displaystyle\frac{1}{2}y+2 \right)\left(2x+y-4\right)$

process
$\displaystyle\frac{2x+y+4}{2}\left(2x+y-4\right)\\
=\displaystyle\frac{(2x+y+4)(2x+y-4)}{2}\\
=\displaystyle\frac{(2x+y)^2-4^2}{2}=\displaystyle\frac{4x^2+4xy+y^2-16}{2}$

$2x^2+2xy+\displaystyle\frac{1}{2}y^2-8$

̎WJȂB\\
$\left( x-\displaystyle\frac{1}{2}y+2 \right)\left(2x+y-4\right)$

process
$\displaystyle\frac{2x-y+4}{2}\left(2x+y-4\right)\\
=\displaystyle\frac{\{2x-(y-4)\}\{2x+(y-4)\}}{2}\\
=\displaystyle\frac{(2x)^2-(y-4)^2}{2}=\displaystyle\frac{4x^2-(y^2-8y+16)}{2}\\
=\displaystyle\frac{4x^2-y^2+8y-16}{2}=2x^2-\displaystyle\frac{1}{2}y^2+4y-8$

$2x^2-\displaystyle\frac{1}{2}y^2+4y-8$

̎WJȂB\\
$(x-y+1)(x-y-1)$

process
$(x-y)^2-1^2$

$x^2-2xy+y^2-1$

̎WJȂB\\
$(x-y+5)(x-y+3)$

process
$(A+5)(A+3)=A^2+8A+15=(x-y)^2+8(x-y)+15$\\
$=x^2-2xy+y^2+8x-8y+15$

$x^2-2xy+y^2+8x-8y+15$

$(a+b+3)(a+b-3)$WJƂǂ̎ɂȂ邩B\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$a^2+b^2+9$\\
\textcircled{\small 2}\,$a^2+b^2-9$\\
\textcircled{\small 3}\,$a^2+2ab+b^2+9$\\
\textcircled{\small 4}\,$a^2+2ab+b^2-9$

process
$a+b=x$ƂƁC\\
$(x+3)(x-3)=x^2-9=(a+b)^2-9=a^2+2ab+b^2-9$

\textcircled{\small 4}

$(x-2y+1)(x+2y+1)$WJ\fbox{\bf \,G\,}ɂȂB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$x^2-4y^2+2x+1$~~~
\textcircled{\small 2}\,$x^2-4y^2+2x-1$\\
\textcircled{\small 3}\,$x^2+4y^2+2x+1$~~~
\textcircled{\small 4}\,$x^2-4y^2+2x-1$

process
$(x+1)^2-4y^2=x^2+2x+1-4y^2$

\textcircled{\small 1}

$(x-3y+2)(x-3y-2)$WJ\fbox{\bf \,I\,}ɂȂB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$x^2+9y^2-4$~~~
\textcircled{\small 2}\,$x^2+9y^2-6xy-4$\\
\textcircled{\small 3}\,$x^2-9y^2+6xy-4$~~~
\textcircled{\small 4}\,$x^2-9y^2-6xy-4$

process
$(x-3y)^2-4=x^2-6xy+9y^2-4$

\textcircled{\small 2}

̎WJȂB\\
$(x-y-2)(x-y+2)$

process
$(x-y)^2-4=x^2-2xy+y^2-4$

$x^2-2xy+y^2-4$

$(x+y+1)^2$WJ\fbox{\hspace{4pt} G \hspace{4pt}}ɂȂB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂
IׁB\\
\textcircled{\small 1}\,$x^2+y^2+1$\\
\textcircled{\small 2}\,$x^2+2xy+y^2+1$\\
\textcircled{\small 3}\,$x^2+xy+y^2+x+y+1$\\
\textcircled{\small 4}\,$x^2+2xy+y^2+2x+2y+1$

process
$(a+1)^2=(x+y)^2+2(x+y)+1$

\textcircled{\small 4}







[Level6]
ibj
̎ȂB\\
$2x^2+5x-3$

$(2x-1)(x+3)$

̎ȂB\\
$3x^2-2x-5$

$(3x-5)(x+1)$

̎ȂB\\
$6x^2+x-2$

$(3x+2)(2x-1)$

̎ȂB\\
$x^2+8x+16$

$(x+4)^2$

̎ȂB\\
$9x^2-12x+4$

$(3x-2)^2$

̎ȂB\\
$x^2+2x-48$

$(x+8)(x-6)$

̎ȂB\\
$x^2-12x+11$

$(x-11)(x-1)$

̎ȂB\\
$3x^2+13x-10$

$(x+5)(3x-2)$

̎ȂB\\
$3x^2+11x-4$

$(x+4)(3x-1)$

̎ȂB\\
$7x^2-15x+2$

$(x-2)(7x-1)$

̎ȂB\\
$3x^2-4x-7$

$(3x-7)(x+1)$

̎ȂB\\
$x^2+5x+4$

$(x+4)(x+1)$

̎ȂB\\
$x^2+8x-20$

$(x-2)(x+10)$

̎ȂB\\
$x^2-9x+18$

$(x-3)(x-6)$

̎ȂB\\
$8x^2-2x-15$

$(2x-3)(4x+5)$

̎ȂB\\
$6x^2-5x+1$

$(2x-1)(3x-1)$

̎ȂB\\
$10x^2+9x-7$

$(5x+7)(2x-1)$

̎ȂB\\
$5x^2-11x+2$

$(5x-1)(x-2)$

̎ȂB\\
$3x^2+11x+6$

$(3x+2)(x+3)$

̎ȂB\\
$3x^2+8x+5$

$(x+1)(3x+5)$

̎ȂB\\
$6x^2-13x-28$

$(3x+4)(2x-7)$

̎ȂB\\
$3x^2+4x+1$

$(3x+1)(x+1)$

̎ȂB\\
$2x^2-15x+7$

$(2x-1)(x-7)$

̎ȂB\\
$3x^2-8x-3$

$(3x+1)(x-3)$

̎ȂB\\
$3x^2-8x-5$

$(3x-5)(x-1)$

̎ȂB\\
$6x^2+19xy+10y^2$

$(3x+2y)(2x+5y)$

̎B\\
$9x^2-4y^2$

process
$a^2-b^2=(a-b)(a+b)$\\
^$=(3x)^2-(2y)^2=(3x-2y)(3x+2y)$

$(3x-2y)(3x+2y)$

̎B\\
$x^2-2x-15$

$(x+3)(x-5)$

̎B\\
$x^2-4x+4$

$(x-2)^2$

̎B\\
$4x^2-8x+3$

$(2x-1)(2x-3)$

̎B\\
$35x^2-8xy-16y^2$

$(5x-4y)(7x+4y)$

̎B\\
$4x^2-4x-3$

$(2x+1)(2x-3)$

$3x^2+7x+2$\\
$\left( x+\fbox{\bf\, A \,} \right) \left( \fbox{\bf\, C \,}x+\fbox{\bf\, E \,} \right) $
ɂȂB

$(x+2)(3x+1)$













[Level7]
ipj
̎ȂB\\
$ab+a+b+1$

process
$a$Ő\\
$a(b+1)+b+1=(b+1)(a+1)$

$(a+1)(b+1)$

̎ȂB\\
$b^2+ab-a-b$

process
ł̒Ⴂ$a$Ő\\
$ab-a+b^2-b=a(b-1)+b(b-1)$

$(a+b)(b-1)$

̎ȂB\\
$(a-b)^2-a+b-2$

process
$(a-b)^2-(a-b)-2=x^2-x-2$\\
$=(x-2)(x+1)=(a-b-2)(a-b+1)$

$(a-b-2)(a-b+1)$

̎ȂB\\
$(a-b)^2-6(a-b)+8$

$(a-b-2)(a-b-4)$

̎ȂB\\
$(x+y)^2+8(x+y)+16$

$(x+y+4)^2$

̎ȂB\\
$(x-2y)^2-16$

$(x-2y-4)(x-2y+4)$

̎ȂB\\
$(a+b)^2-c^2$

$(a+b-c)(a+b+c)$

̎ȂB\\
$ax+bx+a+b$

process
$x$ŐƁC$(a+b)x+a+b$

$(a+b)(x+1)$

̎ȂB\\
$xy+2x-y-2$

$(x-1)(y+2)$

̎ȂB\\
$(x+y)^2-8(x+y)+15$

$(x+y-5)(x+y-3)$

̎ȂB\\
$ac-2bc+a-2b$

process
$a$ŐƁC$a(c+1)-2b(c+1)$

$(a-2b)(c+1)$

̎ȂB\\
$9x^2y+6xy^2-3xy$

$3xy(3x+2y-1)$

̎ȂB\\
$25x^2-9y^2$

$(5x-3y)(5x+3y)$

̎ȂB\\
$2x^3-18x^2+16x$

process
$2x(x^2-9x+8)$

$2x(x-1)(x-8)$

̎ȂB\\
$4a^2bc-8ab^2c-6abc^2$

$2abc(2a-4b-3c)$

̎ȂB\\
$6a^3b^2c-8ab^3c^2$

$2ab^2c(3a^2-4bc)$

̎ȂB\\
$xy+2x-2y-4$

process
$x(y+2)-2(y+2)$

$(x-2)(y+2)$

̎ȂB\\
$ax-bx+by-ay$

process
$(a-b)x+(b-a)y=(a-b)x-(a-b)y$

$(a-b)(x-y)$

̎ȂB\\
$a^3+b^3+a^2b+ab^2$

process
$a$ŐƁC\\
$a^3+ba^2+b^2a+b^3=a^2(a+b)+b^2(a+b)$

$(a^2+b^2)(a+b)$

̎ȂB\\
$4x^2-(x+1)^2$

process
$(2x)^2-(x+1)^2=\{2x-(x+1) \}\{2x+(x+1) \}$

$(x-1)(3x+1)$

̎ȂB\\
$x^4+2x^2y^2+y^4$

process
$(x^2)^2+2x^2y^2+(y^2)^2$

$(x^2+y^2)^2$

̎ȂB\\
$x^2y^2-10xy+25$

$(xy-5)^2$

̎ȂB\\
$a^3-4a^2+4a$

process
$a(a^2-4a+4)$

$a(a-2)^2$

̎ȂB\\
$x^4+2x^3+x^2$

process
$x^2(x^2+2x+1)$

$x^2(x+1)^2$

̎ȂB\\
$y(x+1)-2x-2$

process
$y(x+1)-2(x+1)$

$(x+1)(y-2)$

̎ȂB\\
$ax-3bx+5cx$

$x(a-3b+5c)$

̎ȂB\\
$3a^4-9a^3+15a^2$

$3a^2(a^2-3a+5)$

̎ȂB\\
$3x(2y-1)+5y(2y-1)$

$(2y-1)(3x+5y)$

̎ȂB\\
$a(x-y)-x+y$

$(x-y)(a-1)$

̎ȂB\\
$x^2+x+\displaystyle \frac{1}{4}$

$\left( x+ \displaystyle \frac{1}{2} \right)^2$

̎ȂB\\
$a^2-\displaystyle \frac{2}{3} ab+ \displaystyle \frac{1}{9} b^2$

$\left( a- \displaystyle \frac{1}{3}b \right)^2$

̎ȂB\\
$49x^2-36y^2$

$(7x+6y)(7x-6y)$

̎ȂB\\
$-64+25a^2$

$(5a+8)(5a-8)$

̎ȂB\\
$6x^2+5xy-6y^2$

$(3x-2y)(2x+3y)$

̎ȂB\\
$4x^2-3xy-10y^2$

$(4x+5y)(x-2y)$

̎ȂB\\
$(x+7)^2-(x+7)-6$

process
$x+7=a$ƂƁC\\
$a^2-a-6=(a-3)(a+2)\\
=(x+7-3)(x+7+2)=(x+4)(x+9)$

$(x+4)(x+9)$

̎ȂB\\
$(x+2)^2-6(x+2)-16$

process
$x+2=a$ƂƁC\\
$a^2-6a-16=(a+2)(a-8)\\
=(x+2+2)(x+2-8)=(x+4)(x-6)$

$(x+4)(x-6)$

̎ȂB\\
$3x^2y-6xy$

$3xy(x-2)$

̎ȂB\\
$a(x-y)-b(y-x)$

process
$a(x-y)-b \times(-1)(x-y)\\
=a(x-y)+b(x-y)$

$(x-y)(a+b)$

̎ȂB\\
$x(y-1)-y+1$

process
$x(y-1)-(y-1)=(x-1)(y-1)$

$(x-1)(y-1)$

̎ȂB\\
$abx^2-a^3b$

process
$ab(x^2-a^2)=ab(x-a)(x+a)$

$ab(x-a)(x+a)$

̎ȂB\\
$(x-y)(x-y-3)+2$

process
$x-y=A$ƂƁC\\
$A(A-3)+2=A^2-3A+2=(A-1)(A-2)\\
=(x-y-1)(x-y-2)$

$(x-y-1)(x-y-2)$

̎ȂB\\
$(x^2+3x-2)(x^2+3x+4)-16$

process
$x^2+3x=A$ƂƁC\\
$(A-2)(A+4)-16=A^2+2A-8-16\\
=A^2+2A-24=(A-4)(A+6)\\
=(x^2+3x-4)(x^2+3x+6)\\
=(x+4)(x-1)(x^2+3x+6)$

$(x+4)(x-1)(x^2+3x+6)$

̎ȂB\\
$x^2+xy-yz-zx$

process
$x(x+y)-z(y+x)=(y+x)(x-z)$

$(x+y)(x-z)$

̎ȂB\\
$x^2y+y^2z-y^3-x^2z$

process
$x^2(y-z)+y^2(z-y)\\
=x^2(y-z)-y^2(y-z)=(y-z)(x^2-y^2)\\
=(y-z)(x-y)(x+y)$

$(y-z)(x-y)(x+y)$

̎ȂB\\
$(x-1)^2-2(x-1)-8$

process
$\{ (x-1)+2 \}\{ (x-1)-4 \}$

$(x+1)(x-5)$

̎ȂB\\
$4x^2y^2-2xy^3$

$2xy^2(2x-y)$

̎ȂB\\
$3x^3y^2-9x^2y^4$

$3x^2y^2(x-3y^2)$

̎ȂB\\
$36x^2-81y^2$

$(6x-9y)(6x+9y)$

̎ȂB\\
$2x^2+6ax-8bx$

$2x(x+3a-4b)$

̎ȂB\\
$9x^2+30xy+25y^2$

$(3x+5y)^2$

̎ȂB\\
$(a-b)^2-(a-b)$

$(a-b)(a-b-1)$

̎ȂB\\
$a^2b-ab^2+ab$

$ab(a-b+1)$

̎ȂB\\
$(x-y)^2-7(x-y)+12$

$(x-y-3)(x-y-4)$

̎ȂB\\
$x(y-z)-2y+2z$

process
$x(y-z)-2(y-z)$

$(y-z)(x-2)$

̎ȂB\\
$ab-b-a+1$

process
$b(a-1)-(a-1)$

$(a-1)(b-1)$

̎ȂB\\
$(a-b)xy+(b-a)yz$

process
$(a-b)xy-(a-b)yz$

$y(a-b)(x-z)$

̎ȂB\\
$x^4-2x^2+1$

$(x-1)^2(x+1)^2$

̎ȂB\\
$x^3-x^2-x+1$

process
$x^2(x-1)-(x-1)=(x-1)(x^2-1)$

$(x-1)^2(x+1)$

̎ȂB\\
$x^2-5xy-24y^2$

$(x-8y)(x+3y)$

̎ȂB\\
$1-x+y-xy$

$(1-x)(1+y)$

̎ȂB\\
$(2x+1)^2-(2x+1)-6$

process
$a^2-a-6=(a+2)(a-3)=(2x+1+2)(2x+1-3)$

$2(2x+3)(x-1)$

̎B\\
$2xy-4xz$

$2x(y-2z)$

̎B\\
$2x^2y-xy^2$

$xy(2x-y)$

̎B\\
$(a-b)x+(b-a)y$

$(a-b)(x-y)$

$(x-3)^2+4(x-3)$ȂB

process
$A^2+4A=A(A+4)=(x-3)(x-3+4)=(x-3)(x+1)$

$(x-3)(x+1)$

̎ȂB\\
$x^4-5x^2+4$

process
$(x^2-1)(x^2-4)=(x-1)(x+1)(x-2)(x+2)$

$(x-1)(x+1)(x-2)(x+2)$

̎ȂB\\
$2x^4+x^2 y^2-3y^2$

process
$(2x^2+3y^2)(x^2-y^2)=(2x^2+3y^2)(x-y)(x+y)$

$(2x^2+3y^2)(x-y)(x+y)$

̎ȂB\\
$x^2+9y^2-25z^2-6xy$

process
$x^2+9y^2-25z^2-6xy=(x-3y)^2-25z^2$

$(x-3y-5z)(x-3y+5z)$







[Level8]
̒lCΏ̎
$x=2Cy=-4$̂ƂC\\
$5(3x+2y)-3(4x+3y)$̒l߂ȂB

process
$5(3x+2y)-3(4x+3y)=3x+y=3 \times 2 +(-4)=2$

2

$x=-2Cy=5$̂ƂC\\
$4x^2y^3 \div 8xy^2 \times 6x$̒l߂ȂB

process
$4x^2y^3 \div 8xy^2 \times 6x=3x^2y=3(-2)^2 \times 5$

60

$a=\displaystyle \frac{1}{2}Cb=-5$̂ƂC\\
$3(a+b)-(a+4b)$̒l߂ȂB

process
$3(a+b)-(a+4b)=2a-b=2 \times \displaystyle \frac{1}{2}-(-5)$

6

$a=2Cb=-3$̂ƂC$3(2a-3b)+2(4a+b)$̒l߂B

process
$14a-7b=14 \times 2 -7 \times (-3)=49$

49

$x=8Cy=-6$̂ƂC$3x \div 4y$̒l߂B

process
$\displaystyle \frac{3 \times 8}{4 \times (-2)}$

$-1$

$x=3Cy=-2$̂ƂC$12xy \div 3 \times 2y$̒l߂B

process
$8xy^2=8 \times 3 \times (-2)^2=96$

96

$a=-2$C$b=3$̂ƂC$\displaystyle \frac{b}{a}-\displaystyle \frac{ab}{b^2}$̒l߂B

process
$\displaystyle \frac{3}{-2}-\displaystyle \frac{-2 \times 3}{3^2}=-\displaystyle \frac{3}{2}+\displaystyle \frac{2}{3}
=-\displaystyle \frac{9}{6}+\displaystyle \frac{4}{6}=-\displaystyle \frac{5}{6}$

$-\displaystyle \frac{5}{6}$

$a=\displaystyle \frac{1}{2}$C$b=4$̂ƂC$3 \left( 2a-3b \right)-2\left( -3a-2b \right)$̒l߂B

process
$12a-5b=12 \times \displaystyle \frac{1}{2} -5 \times 4= 6-20=-14$

$-14$

$a=\displaystyle \frac{4}{5}$C$b=\displaystyle \frac{1}{3}$̂ƂC$\left( 5a^2-3ab^2 \right) \div ab$̒l߂B

process
$\displaystyle \frac{5a^2-3ab^2}{ab}=5a-3b=5 \times \displaystyle \frac{4}{5}-3 \times \displaystyle \frac{1}{3}=4-1=3$

$3$

$x=2$C$y=\displaystyle \frac{1}{4}$̂ƂC$-4x^2y^2 \div 2xy^2 \times 3x^2y$̒l߂B

process
$-\displaystyle \frac{4x^2y^2}{2xy^2} \times 3x^2y=-6x^3y=-6 \times 2^3 \times \displaystyle \frac{1}{4}\\
=-6 \times 8 \times \displaystyle \frac{1}{4}=-12 $

$-12$

$x=-3$C$y=5$̂ƂC$\left( 2x+5y \right)\left( 2x-5y \right)-\left( x-4y \right)\left( x+6y \right)$̒l߂B

process
$4x^2-25y^2-\left( x^2+2yx-24y^2 \right)\\
=4x^2-25y^2-x^2-2yx+24y^2\\
=3 x^2-2yx-y^2=3 \times (-3)^2-5^2-2 \times 5 \times (-3)\\
=3 \times 9 -25 +30=32$

$32$

$x=1+\sqrt{5},y=1-\sqrt{5}$̂ƂC$x^2-y^2$̒l߂ȂB

process
$x^2-y^2=(x+y)(x-y)=2 \times 2\sqrt{5}=4 \sqrt{5}$

$4 \sqrt{5}$

$a=\sqrt{5}$A$b=\sqrt{3}$̂ƂA$(a-b)^2+2ab$̒l߂B

process
$(a-b)^2+2ab=a^2-2ab+b^2+2ab=a^2+b^2
=\left( \sqrt{5} \right)^2+\left( \sqrt{3} \right)^2=5+3=8$

8

$x+y=-4$C$xy=3$̂ƂC̊e̒l߂ȂB\\
\textcircled{\small 1}$x^2+y^2$\\
\textcircled{\small 2}$\displaystyle \frac{1}{x}+\displaystyle \frac{1}{y}$

process
\textcircled{\small 1}$x^2+y^2=(x+y)^2-2xy=(-4)^2-2 \times 3\\
=16-6=10$\\
\textcircled{\small 2}$\displaystyle \frac{1}{x}+\displaystyle \frac{1}{y}=\displaystyle \frac{y+x}{xy}=\displaystyle \frac{-4}{3}$

\textcircled{\small 1}$10$C\textcircled{\small 2}$-\displaystyle \frac{4}{3}$

$x=1+\sqrt{2}$C$y=1-\sqrt{2}$̂ƂC$x^2+y^2$̒l߂B

process
$x^2+y^2=(1+\sqrt{2})^2+(1-\sqrt{2})\\
=1+2+2\sqrt{2}+1+2-2\sqrt{2}=6$\\
iʉj$x+y=2$C$xy=1^2-(\sqrt{2})^2=1-2=-1$\\
$x^2+y^2=(x+y)^2-2xy=2^2-2 \times (-1)=6$

$6$




[EOF]
