[Title]
201(2)_ƕs
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% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
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% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB

[Level1]
2i\bj
̕ȂB\\
$x\left( x-3 \right)=5\left( x-3 \right)$

process
$x^2-3x=5x-15$, $x^2-8x+15=0$, \\
$(x-3)(x-5)=0$,\,$x=3, 5$

$x=3, 5$

̕ȂB\\
$x^2+\displaystyle \frac{7}{3} x+\displaystyle \frac{2}{3}=0$

process
$3x^2+7x+2=0$,$(3x+1)(x+2)=0$,$x=-2,\, -\displaystyle \frac{1}{3}$

$x=-2,\, -\displaystyle \frac{1}{3}$

2B\\
$x^2-6x+8=0$

process
$(x-2)(x-4)=0$

$x=2,4$

2B\\
$x^2+4x+4=0$

process
$(x+2)^2=0$

$x=-2$

2B\\
$2x^2-x-3=0$

process
$(2x-3)(x+1)=0$

$x=\displaystyle \frac{3}{2},-1$

2B\\
$x^2-6x-91=0$

process
$(x+7)(x-13)=0$

$x=-7,13$

2$5x^2+7x-6=0$ȂB

process
$(5x-3)(x+2)=0$

$\displaystyle \frac{3}{5}C-2$




[Level2]
2i\pj
̕ȂB\\
$3(2x+3)(2x-1)=-(2x-3)^2+4$

process
$3(4x^2+2\times 2x-3)=-(4x^2-12x+9)+4$\\
$12x^2+12x-9=-4x^2+12x-9+4$,\\
$16x^2=4$,\,$x=\pm \displaystyle \frac{1}{2}$

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

鐮$x$32悵̂́C
$x$3{12{ɓB
𖞂2̐$x$̂C傫̐߂B

process
$(x-3)^2=2(3x-1)$C$x^2-6x+9=6x-2$\\
$(x-1)(x-11)=0$C$x=1,11$

11

̕ȂB\\
$(x+2)(x-3)=2(x^2-4)$

process
$x^2-x-6=2x^2-8$,\, $x^2+x-2=0$,\\
$(x+2)(x-1)=0$

$x=-2,1$

̕ȂB\\
$(x+5)^2-2(x+5)-8=0$

process
$x+5=A$ƂƁC$A^2-2A-8=0$,\\
$(A+2)(A-4)=0$,\, $A=-2,\,4$,\, $x+5=-2,\,4$

$x=-7,-1$









[Level3]
2is\j
2B\\
$(x-1)^2=5$

process
$x-1= \pm \sqrt{5}$

$x=1 \pm \sqrt{5}$

2B\\
$x^2-4x+2=0$\\

process
$x^2-4x=-2$\\
$(x-2)^2=-2+4$\\
$(x-2)^2=2$\\
$x-2=\pm \sqrt{2}$

$x=2\pm \sqrt{2}$

2B\\
$x^2-4x-3=0$\\

process
$x^2-4x=3$\\
$(x-2)^2=3+4$\\
$(x-2)^2=7$\\
$x-2=\pm \sqrt{7}$

$x=2\pm \sqrt{7}$

2B\\
$x^2-4x-1=0$\\

process
$x^2-4x=1$\\
$(x-2)^2=1+4$\\
$(x-2)^2=5$\\
$x-2=\pm \sqrt{5}$

$x=2\pm \sqrt{5}$

2B\\
$x^2-4x+1=0$\\

process
$x^2-4x=-1$\\
$(x-2)^2=-1+4$\\
$(x-2)^2=3$\\
$x-2=\pm \sqrt{3}$

$x=2\pm \sqrt{3}$

2B\\
$x^2-2x-2=0$

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

$x=1\pm \sqrt{3}$

2B\\
$x^2-2x-4=0$

process
$x^2-2x=4$\\
$x^2-2x+1=4+1$\\
$(x-1)^2=5$ $\therefore x-1=\pm \sqrt{5}$

$x=1\pm \sqrt{5}$

2B\\
$x^2-2x-6=0$

process
$x^2-2x=6$\\
$x^2-2x+1=6+1$\\
$(x-1)^2=7$ $\therefore x-1=\pm \sqrt{7}$

$x=1\pm \sqrt{7}$

$x^2+2x-1=0$

process
$x^2+2x=1$\\
$x^2+2x+1=1+1$\\
$(x+1)^2=2$ $\therefore x+1=\pm \sqrt{2}$

$x=-1\pm \sqrt{2}$

$x^2+2x-2=0$

process
$x^2+2x=2$\\
$x^2+2x+1=2+1$\\
$(x+1)^2=3$ $\therefore x+1=\pm \sqrt{3}$

$x=-1\pm \sqrt{3}$

$x^2+2x-4=0$

process
$x^2+2x=4$\\
$x^2+2x+1=4+1$\\
$(x+1)^2=5$ $\therefore x+1=\pm \sqrt{5}$

$x=-1\pm \sqrt{5}$

$x^2+2x-6=0$

process
$x^2+2x=6$\\
$x^2+2x+1=6+1$\\
$(x+1)^2=7$ $\therefore x+1=\pm \sqrt{7}$

$x=-1\pm \sqrt{7}$

$x^2+4x+2=0$

process
$x^2+4x=-2$\\
$x^2+4x+4=-2+4$\\
$(x+2)^2=2$ $\therefore x+2=\pm \sqrt{2}$

$x=-2\pm \sqrt{2}$

$x^2+4x+1=0$

process
$x^2+4x=-1$\\
$x^2+4x+4=-1+4$\\
$(x+2)^2=3$ $\therefore x+2=\pm \sqrt{3}$

$x=-2\pm \sqrt{3}$

$x^2+4x-1=0$

process
$x^2+4x=1$\\
$x^2+4x+4=1+4$\\
$(x+2)^2=5$ $\therefore x+2=\pm \sqrt{5}$

$x=-2\pm \sqrt{5}$

$x^2+4x-3=0$

process
$x^2+4x=3$\\
$x^2+4x+4=3+4$\\
$(x+2)^2=7$ $\therefore x+2=\pm \sqrt{7}$

$x=-2\pm \sqrt{7}$

2B\\
$3x^2-4=0$

process
$x^2=\displaystyle \frac{4}{3}$C
$x=\pm \displaystyle \frac{2}{\sqrt{3}}=\pm \displaystyle \frac{2\sqrt{3}}{3}$

$x=\pm \displaystyle \frac{2\sqrt{3}}{3}$

2B\\
$x^2+6x+3=0$\\

process
$x^2+6x=-3$\\
$(x+3)^2=-3+9$\\
$(x+3)^2=6$\\
$x+3=\pm \sqrt{6}$

$x=-3\pm \sqrt{6}$









[Level4]
2ǐpj
̕ȂB\\
$\displaystyle \frac{x^2-2}{3}-\displaystyle \frac{x^2-1}{2}=-2x$

process
ӂ6ƁC\\
$2(x^2-2)-3(x^2-1)=-12x$,\\
$2x^2-4-3x^2+3=-12x$,\,$x^2-12+1=0$\\
̌C\\
$x=\displaystyle \frac{12 \pm \sqrt{(-12)^2-4 \times 1 \times 1}}{2 \times 1}
=\displaystyle \frac{12 \pm \sqrt{140}}{2}\\
=\displaystyle \frac{12 \pm 2\sqrt{35}}{2}
=6 \pm \sqrt{35}$

$6 \pm \sqrt{35}$

2$4x^2-5x-1=0$ȂB

process
$x=\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}\\
=\displaystyle \frac{-(-5)\pm \sqrt{25-4 \times 4 \times (-1)}}{2 \times 4}
=\displaystyle \frac{5 \pm \sqrt{41}}{8}$

$\displaystyle \frac{5 \pm \sqrt{41}}{8}$

2$x^2+9x+5=0$ȂB

process
$a=1,b=9,c=5$\\
$x=\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}\\
=\displaystyle \frac{-9\pm \sqrt{9^2-4 \times 1 \times 5}}{2 \times 1}
=\displaystyle \frac{-9 \pm \sqrt{61}}{2}$

$\displaystyle \frac{-9 \pm \sqrt{61}}{2}$

2$x^2+5x-7=0$̉߂B

process
$a=1,b=5,c=-7$\\
$x=\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}\\
=\displaystyle \frac{-5 \pm \sqrt{25-4 \times 1 \times (-7)}}{2 \times 1}
=\displaystyle \frac{-5 \pm \sqrt{53}}{2}$

$\displaystyle \frac{-5 \pm \sqrt{53}}{2}$

2$3x^2+x-2=0$ȂB

process
$(3x-2)(x+1)=0$

$x=\displaystyle \frac{2}{3},-1$

2$x^2-5x-7=0$̉߂B

process
$a=1,b=-5,c=-7$\\
$x=\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}\\
=\displaystyle \frac{5 \pm \sqrt{25-4 \times 1 \times (-7)}}{2 \times 1}
=\displaystyle \frac{5 \pm \sqrt{53}}{2}$

$\displaystyle \frac{5 \pm \sqrt{53}}{2}$

Q$2x^2+x-4=0$ȂB

process
$a=2$C$b=1$C$c=-4$ƂāC\\
$x=\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}\\
=\displaystyle \frac{-1\pm \sqrt{1^2-4 \times 2 \times (-4)}}{2 \times 2}
=\displaystyle \frac{-1 \pm \sqrt{33}}{4}$

$\displaystyle \frac{-1 \pm \sqrt{33}}{4}$

2B\\
$3x^2-5x-4=0$

process
$x=\displaystyle\frac{5\pm \sqrt{(-5)^2-4\times 3 \times(-4)}}{2\times 3}$

$x=\displaystyle\frac{5 \pm \sqrt{73}}{6}$

2B\\
$x^2-7x+5=0$

process
$x=\displaystyle\frac{7\pm \sqrt{(-7)^2-4\times 1 \times 5}}{2\times 1}$

$x=\displaystyle\frac{7 \pm \sqrt{29}}{2}$

2B\\
$2x^2-x-7=0$

process
$x=\displaystyle\frac{7\pm \sqrt{(-1)^2-4\times 2 \times (-7)}}{2\times 2}$

$x=\displaystyle\frac{7 \pm \sqrt{29}}{2}$

2ȂB\\
$2x^2+3x-1=0$

process
$x=\displaystyle\frac{-3\pm\sqrt{3^2-4\times 2 \times (-1)}}{2\times 2}$

$x=\displaystyle\frac{-3\pm\sqrt{17}}{4}$

2ȂB\\
$3x^2-7x+1=0$

process
$x=\displaystyle\frac{7\pm\sqrt{(-7)^2-4\times 3 \times 1}}{2\times 3}$

$x=\displaystyle\frac{7\pm\sqrt{37}}{6}$

2ȂB\\
$2x^2-4x-3=0$

process
$b'=-2$\\
$x=\displaystyle\frac{-b'\pm\sqrt{{b'}^2-ac}}{a}
=\displaystyle\frac{2\pm\sqrt{(-2)^2-2 \times (-3)}}{2}$

$x=\displaystyle\frac{2\pm\sqrt{10}}{2}$

2ȂB\\
$3x^2-x-3=0$

process
$x=\displaystyle\frac{1\pm\sqrt{(-1)^2-4\times 3 \times (-3)}}{2\times 3}$

$x=\displaystyle\frac{1\pm\sqrt{37}}{6}$

2ȂB\\
$3x^2-6x+1=0$

process
$b'=-3$\\
$x=\displaystyle\frac{-b'\pm\sqrt{{b'}^2-ac}}{a}
=\displaystyle\frac{3\pm\sqrt{(-3)^2-3 \times 1}}{3}$

$x=\displaystyle\frac{3\pm\sqrt{6}}{3}$

2ȂB\\
$5x^2-x-2=0$

process
$x=\displaystyle\frac{1\pm\sqrt{(-1)^2-4\times 5 \times (-2)}}{2\times 5}$

$x=\displaystyle\frac{1\pm\sqrt{41}}{10}$









[Level5]
QƉ
$x$2$x^2-kx-2k-10=0$i$k$͒萔j̉̈$k$łƂC
$k$̒l߂B

process
$k^2-k^2-2k-10=0$,\  $-2k-10=0$\\
$k=-5$

$-5$

̕$x$̒l$-3$̂ƂCA̒l߂B\\
$3x^2+2Ax-19+A^2=0$

process
^$x=-3$ƁC\\
$3(-3)^2+2A\times(-3)-19+A^2=0$\\
$3 \times 9 -6A-19+A^2=0$\\
$A^2-6A+8=0$,\,$(A-2)(A-4)=0$,\,$A=2,\,4$

$A=2,\,4$

2$x^2+ax+b=0$2̉$-2,\, -5$łƂC$a, \,b$̒l߂B

process
ӂC2́C$k(x+2)(x+5)=0$i$k$͒萔jƂƂłB
̎WJƁC$k(x^2+7x+10)=0$B^$x^2$̌W1ł邩C$k=1$B

$a=7,\,b=10$

$x$2$2x^2+mx+m+7=0im͒萔j$̉̈2łƂC
$m$̒l߂B

process
$2 \times 2^2+m \times 2+m+7=0$\\
$8+2m+m+7=0$,\ $m=-5$

$-5$

$x$2 $2x^{2}-kx+2k+4=0 ik͒萔j$̉̈$-2$łƂC$k$̒l߂B

process
$x$$-2$\\
$8+2k+2k+4=0$

$k=-3$

$x$2 $x^{2}-kx+3k-2=0$ i$k$͒萔j̉̈2łƂC$k$̒l߂B

process
$x$2\\
$4-2k+3k-2=0$

$k=-2$

$x$2 $4x^{2}-3mx-m=0$im͒萔j̉̈$-1$łƂC$m$̒l߂B

process
$x$$-1$\\
$4+3m-m=0$

$m=-2$

$x$2 $x^{2}-kx-2k-10=0$ i$k$͒萔j̉̈$k$łƂC$k$̒l߂B

process
$x$$k$\\
$k^{2}-k^{2}-2k-10=0$\\

$k=-5$

$x$2 $x^{2}+(a-5)x-6=0$ i$a$͒萔j̉̈3łƂC$a$̒lƂ̉߂B

process
$x$3\\
$9+3(a-5)-6=0$\\
$a=4$\\
$a$̒l^ɑƁC\\
$x^{2}-x-6=0$\\
$(x-3)(x+2)=0$

$a=4Cx=-2$

$x$2$2x^2+(3k-1)x-4k=0$i$k$͒萔j̉̈2łƂC
$k$̒lƂ̉߂B

process
$x=2$ƁC\\
$2 \times 2^2+(3k-1)\times 2-4k=0$C$k=-3$\\
$k=-3$^ɑƁC
$2x^2-10x+12=0$C$(x-2)(x-3)=0$

$k=-3,\, x=3$

$x$2$x^2-(k+5)x+3k^2=0$i$k$͒萔j̉̈3łƂC
$k$̒l߂B

process
$x=3$ƁC\\
$3^2-(k+5)\times 3+3k^2=0$\\
$k^2-k-2=0$C$(k+1)(k-2)=0$

$k=-1,\,2$

2
$3x^2+5x+k=0$قȂ2̎ƂC
$k$̒l͈̔͂߂B

process
ӂ
$25-12k>0 \therefore k<\displaystyle\frac{25}{12}$

$k<\displaystyle\frac{25}{12}$





[Level6]



[Level7]

[Level8]

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