Question

Tìm x

(2x+5)(4x² -10x+25)-(2x+1)² =52

Answer 1

Ta có: \(\left(2x+5\right)\left(4x^2-10x+25\right)-\left(2x+1\right)^2=52\)

<=> \(8x^3+125-4x^2-4x-1=52\)

<=> \(8x^3-4x^4-4x+72=0\)

<=> \(2x^3-x^2-x+18=0\)

<=> \(\left(2x^3+4x^2\right)-\left(5x^2+10x\right)+\left(9x+18\right)=0\)

<=> \(2x^2\left(x+2\right)-5x\left(x+2\right)+9\left(x+2\right)=0\)

<=> \(\left(x+2\right)\left(2x^2-5x+9\right)=0\)

<=> \(\left[{}\begin{matrix}x+2=0\left(1\right)\\2x^2-5x+9=0\left(2\right)\end{matrix}\right.\)

Mặt khác :

\(2x^2-5x+9=2\left(x^2-2,5x+1,5625\right)+5,875\)

= \(2\left(x-1,25\right)^2+5,875\) \(\ge5,875\) > 0

=> PT (2) vô nghiệm

=> x + 2 = 0 => \(x=-2\)

Vậy nghiệm của PT : S = \(\left\{-2\right\}\)