Solve 5xy^2-25xy+30x

$$5 x y ^ { 2 } - 25 x y + 30 x$$

Factor

$$5x\left(y-3\right)\left(y-2\right)$$

Solution Steps

Factor out 5.

$$5\left(xy^{2}-5xy+6x\right)$$

Consider xy^{2}-5xy+6x. Factor out x.

$$x\left(y^{2}-5y+6\right)$$

Consider y^{2}-5y+6. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 6=6$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -5.

$$a=-3 b=-2$$

Rewrite y^{2}-5y+6 as \left(y^{2}-3y\right)+\left(-2y+6\right).

$$\left(y^{2}-3y\right)+\left(-2y+6\right)$$

Factor out y in the first and -2 in the second group.

$$y\left(y-3\right)-2\left(y-3\right)$$

Factor out common term y-3 by using distributive property.

$$\left(y-3\right)\left(y-2\right)$$

Rewrite the complete factored expression.

$$5x\left(y-3\right)\left(y-2\right)$$

Evaluate

$$5x\left(y-3\right)\left(y-2\right)$$