Question

find all real solutions to the equation. $3x^3 - 2x^2 + 75x - 50 = 0$

Explanation:

Step1: Group polynomial terms

$(3x^3 - 2x^2) + (75x - 50) = 0$

Step2: Factor out GCF from groups

$x^2(3x - 2) + 25(3x - 2) = 0$

Step3: Factor out common binomial

$(3x - 2)(x^2 + 25) = 0$

Step4: Set factors to 0, solve

$3x - 2 = 0 \implies x = \frac{2}{3}$
$x^2 + 25 = 0 \implies x^2 = -25 \implies x = \pm 5i$

Step5: Identify real solutions

Only $x = \frac{2}{3}$ is real.

Answer:

$x = \frac{2}{3}$, $x = 5i$, $x = -5i$; real solution: $x = \frac{2}{3}$