# Violympic toán 9

Giải phương trình 1, $x^2+9x+7=\left(2x+1\right)\sqrt{2x^2+4x+5}$

2, GPT $\left(2x+7\right)\sqrt{2x+7}=x^2+9x+7$

3. GHPT $\left\{{}\begin{matrix}x^2-2y-1=2\sqrt{5y+8}+\sqrt{7x-1}\\\left(x-y\right)\left(x^2+xy+y^2+3\right)=3\left(x^2+y^2\right)+2\end{matrix}\right.$

Trung tá -
14 tháng 1 lúc 13:15

1.

$\Leftrightarrow\left(2x+1\right)\sqrt{2x^2+4x+5}-\left(2x+1\right)\left(x+3\right)+x^2-2x-4=0$

$\Leftrightarrow\left(2x+1\right)\left(\sqrt{2x^2+4x+5}-\left(x+3\right)\right)+x^2-2x-4=0$

$\Leftrightarrow\dfrac{\left(2x+1\right)\left(x^2-2x-4\right)}{\sqrt{2x^2+4x+5}+x+3}+x^2-2x-4=0$

$\Leftrightarrow\left[{}\begin{matrix}x^2-2x-4=0\\\dfrac{2x+1}{\sqrt{2x^2+4x+5}+x+3}+1=0\left(1\right)\end{matrix}\right.$

$\left(1\right)\Leftrightarrow2x+1+\sqrt{2x^2+4x+5}+x+3=0$

$\Leftrightarrow\sqrt{2x^2+4x+5}=-3x-4$ $\left(x\le-\dfrac{4}{3}\right)$

$\Leftrightarrow2x^2+4x+5=9x^2+24x+16$

$\Leftrightarrow7x^2+20x+11=0$

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Trung tá -
14 tháng 1 lúc 13:15

2.

ĐKXĐ: ...

$\Leftrightarrow2x\sqrt{2x+7}+7\sqrt{2x+7}=x^2+2x+7+7x$

$\Leftrightarrow\left(x^2-2x\sqrt{2x+7}+2x+7\right)+7\left(x-\sqrt{2x+7}\right)=0$

$\Leftrightarrow\left(x-\sqrt{2x+7}\right)^2+7\left(x-\sqrt{2x+7}\right)=0$

$\Leftrightarrow\left(x-\sqrt{2x+7}\right)\left(x+7-\sqrt{2x+7}\right)=0$

$\Leftrightarrow\left[{}\begin{matrix}x=\sqrt{2x+7}\\x+7=\sqrt{2x+7}\end{matrix}\right.$

$\Leftrightarrow...$

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Trung tá -
14 tháng 1 lúc 13:21

3.

ĐKXĐ: ...

Từ pt dưới:

$\Leftrightarrow\left(x-y\right)\left(x^2+xy+y^2\right)+3x-3y=3x^2+3y^2+1+1$

$\Leftrightarrow x^3-y^3+3x-3y=3x^2+3y^2+1+1$

$\Leftrightarrow x^3-3x^2+3x-1=y^3+3y^2+3y+1$

$\Leftrightarrow\left(x-1\right)^3=\left(y+1\right)^3$

$\Leftrightarrow y=x-2$

Thế vào pt trên:

$x^2-2x+3=2\sqrt{5x-2}+\sqrt{7x-1}$

$\Leftrightarrow x^2-5x+2+2\left(x-\sqrt{5x-2}\right)+\left(x+1-\sqrt{7x-1}\right)=0$

$\Leftrightarrow x^2-5x+2+\dfrac{2\left(x^2-5x+2\right)}{x+\sqrt{5x-2}}+\dfrac{x^2-5x+2}{x+1+\sqrt{7x-1}}=0$

$\Leftrightarrow x^2-5x+2=0$

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