Question

Giải phương trình: \(\left(x+2\right).\left(x-3\right).\left(x^2+2x-24\right)=16x^2\)

Answer 1

(x + 2)(x - 3)(x² + 2x - 24) = 16x² 

<=> (x + 2)(x - 3)[(x + 1)² - 25] = 16x² 

<=> (x + 2)(x - 3)(x + 6)(x - 4) = 16x² 

<=> (x² + 8x + 12)(x² - 7x + 12) = 16x² 

<=> \(\left(x^2+0,5x+12+7,5x\right)\left(x^2+0,5x+12-7.5x\right)=16x^2\)

<=> \(\left(x^2+0,5x+12\right)^2-\left(7,5x\right)^2=16x^2\)

<=> \(\left(x^2+0,5x+12\right)^2=\left(8,5x\right)^2\)

<=> \(\left(x^2+9x+12\right)\left(x^2-8x+12\right)=0\)

<=> \(\left(x+\dfrac{9}{2}-\dfrac{\sqrt{33}}{2}\right)\left(x+\dfrac{9}{2}+\dfrac{\sqrt{33}}{2}\right)\left(x-2\right)\left(x-6\right)=0\)

<=>\(\left[{}\begin{matrix}x=\dfrac{\sqrt{33}-9}{2}\\x=\dfrac{-\sqrt{33}-9}{2}\\x=2\\x=6\end{matrix}\right.\)