Question

6x^3 + 110x^2 − 200x = 15000 write and solve

Answer 1

Solve 6x^3 + 110x^2 − 200x = 15000.

Bring all terms to one side:
6x^3 + 110x^2 − 200x − 15000 = 0.
Divide by 2:
3x^3 + 55x^2 − 100x − 7500 = 0.

Try x = 10: 3(10)^3 + 55(10)^2 − 100(10) − 7500 = 3000 + 5500 − 1000 − 7500 = 0, so (x − 10) is a factor.
Divide the cubic by (x − 10) (synthetic division) to get:
3x^3 + 55x^2 − 100x − 7500 = (x − 10)(3x^2 + 85x + 750).

Solve 3x^2 + 85x + 750 = 0. Discriminant Δ = 85^2 − 4·3·750 = 7225 − 9000 = −1775 = −25·71.
Thus the quadratic has complex roots:
x = [−85 ± i·√1775]/6 = [−85 ± 5i·√71]/6.

Solutions:
x = 10 (real),
x = (−85 + 5i√71)/6,
x = (−85 − 5i√71)/6.