Solve 6x^2y-28xy+30y

$$6 x ^ { 2 } y - 28 x y + 30 y$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(3x^{2}y-14xy+15y\right)$$

Consider 3x^{2}y-14xy+15y. Factor out y.

$$y\left(3x^{2}-14x+15\right)$$

Consider 3x^{2}-14x+15. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-14 ab=3\times 15=45$$

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 45.

$$-1,-45 -3,-15 -5,-9$$

Calculate the sum for each pair.

$$-1-45=-46 -3-15=-18 -5-9=-14$$

The solution is the pair that gives sum -14.

$$a=-9 b=-5$$

Rewrite 3x^{2}-14x+15 as \left(3x^{2}-9x\right)+\left(-5x+15\right).

$$\left(3x^{2}-9x\right)+\left(-5x+15\right)$$

Factor out 3x in the first and -5 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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