Solve -2y^3+48y^2-190y

$$- 2 y ^ { 3 } + 48 y ^ { 2 } - 190 y$$

Factor

$$2y\left(5-y\right)\left(y-19\right)$$

Solution Steps

Factor out 2.

$$2\left(-y^{3}+24y^{2}-95y\right)$$

Consider -y^{3}+24y^{2}-95y. Factor out y.

$$y\left(-y^{2}+24y-95\right)$$

Consider -y^{2}+24y-95. Factor the expression by grouping. First, the expression needs to be rewritten as -y^{2}+ay+by-95. To find a and b, set up a system to be solved.

$$a+b=24 ab=-\left(-95\right)=95$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 95.

$$1,95 5,19$$

Calculate the sum for each pair.

$$1+95=96 5+19=24$$

The solution is the pair that gives sum 24.

$$a=19 b=5$$

Rewrite -y^{2}+24y-95 as \left(-y^{2}+19y\right)+\left(5y-95\right).

$$\left(-y^{2}+19y\right)+\left(5y-95\right)$$

Factor out -y in the first and 5 in the second group.

$$-y\left(y-19\right)+5\left(y-19\right)$$

Factor out common term y-19 by using distributive property.

$$\left(y-19\right)\left(-y+5\right)$$

Rewrite the complete factored expression.

$$2y\left(y-19\right)\left(-y+5\right)$$

Evaluate

$$2y\left(5-y\right)\left(y-19\right)$$