Exercise Zone : Persamaan Bentuk Akar

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Tipe:

Standar SBMPTN HOTS

No.

Nilai x real yang memenuhi persamaan {\sqrt{x^2-x+1}-\sqrt{x^2+2x+4}=\sqrt{3x^2+2x+1}-\sqrt{3x^2-1}} adalah ....

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\begin{aligned} \sqrt{x^2-x+1}-\sqrt{x^2+2x+4}\cdot\dfrac{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}&=\sqrt{3x^2+2x+1}-\sqrt{3x^2-1}\cdot\dfrac{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}\\[8pt] \dfrac{\left(x^2-x+1\right)-\left(x^2+2x+4\right)}{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}&=\dfrac{\left(3x^2+2x+1\right)-\left(3x^2-1\right)}{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}\\[8pt] \dfrac{x^2-x+1-x^2-2x-4}{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}&=\dfrac{3x^2+2x+1-3x^2+1}{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}\\[8pt] \dfrac{-3x-3}{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}&=\dfrac{2x+2}{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}\\[8pt] \dfrac{-3(x+1)}{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}&=\dfrac{2(x+1)}{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}\\[8pt] \dfrac{2(x+1)}{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}+\dfrac{3(x+1)}{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}&=0\\[8pt] (x+1)\left(\dfrac2{\sqrt{3x^2+2x+1}+\sqrt{3x^2-1}}+\dfrac3{\sqrt{x^2-x+1}+\sqrt{x^2+2x+4}}\right)&=0\\[8pt] x+1&=0\\ x&=\boxed{\boxed{-1}} \end{aligned}

Jadi, x=-1