Solve xy^2-9xy+18x

$$x y ^ { 2 } - 9 x y + 18 x$$

Factor

$$x\left(y-6\right)\left(y-3\right)$$

Solution Steps

Factor out x.

$$x\left(y^{2}-9y+18\right)$$

Consider y^{2}-9y+18. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+18. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\times 18=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -9.

$$a=-6 b=-3$$

Rewrite y^{2}-9y+18 as \left(y^{2}-6y\right)+\left(-3y+18\right).

$$\left(y^{2}-6y\right)+\left(-3y+18\right)$$

Factor out y in the first and -3 in the second group.

$$y\left(y-6\right)-3\left(y-6\right)$$

Factor out common term y-6 by using distributive property.

$$\left(y-6\right)\left(y-3\right)$$

Rewrite the complete factored expression.

$$x\left(y-6\right)\left(y-3\right)$$

Evaluate

$$x\left(y-6\right)\left(y-3\right)$$