Question

find all real solutions to the equation. $16x^3 - 16x^2 - 9x + 9 = 0$

Explanation:

Step1: Group polynomial terms

$(16x^3 - 16x^2) + (-9x + 9) = 0$

Step2: Factor out common terms

$16x^2(x - 1) - 9(x - 1) = 0$

Step3: Factor out shared binomial

$(x - 1)(16x^2 - 9) = 0$

Step4: Factor difference of squares

$(x - 1)(4x - 3)(4x + 3) = 0$

Step5: Solve for x via zero product

$x - 1 = 0 \implies x = 1$
$4x - 3 = 0 \implies x = \frac{3}{4}$
$4x + 3 = 0 \implies x = -\frac{3}{4}$

Answer:

$x = 1$, $x = \frac{3}{4}$, $x = -\frac{3}{4}$