Question

6x^3 + 110x^2 − 200x = 15000 write and solve
𝑦=____𝑥^3+____𝑥^2+___𝑥,
𝑦=___;𝑥= cm

Answer 1

Write y = 6x^3 + 110x^2 − 200x.
Also y = 15000.

Solve 6x^3 + 110x^2 − 200x = 15000:
Subtract 15000: 6x^3 + 110x^2 − 200x − 15000 = 0.
Divide by 2: 3x^3 + 55x^2 − 100x − 7500 = 0.
Test x = 10: 3(10)^3 + 55(10)^2 − 100(10) − 7500 = 0, so (x − 10) is a factor.

Divide: 3x^3 + 55x^2 − 100x − 7500 = (x − 10)(3x^2 + 85x + 750).
The quadratic 3x^2 + 85x + 750 has discriminant 85^2 − 4·3·750 = −1775 < 0, so no real roots.

Therefore the only real solution is x = 10 cm.

Final answers:
y = 6x^3 + 110x^2 − 200x
y = 15000; x = 10 cm.