Solve ax^2-7ax+6a

$$a x ^ { 2 } - 7 a x + 6 a$$

Factor

$$a\left(x-6\right)\left(x-1\right)$$

Solution Steps

Factor out a.

$$a\left(x^{2}-7x+6\right)$$

Consider x^{2}-7x+6. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+px+qx+6. To find p and q, set up a system to be solved.

$$p+q=-7 pq=1\times 6=6$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q 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 -7.

$$p=-6 q=-1$$

Rewrite x^{2}-7x+6 as \left(x^{2}-6x\right)+\left(-x+6\right).

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

Factor out x in the first and -1 in the second group.

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

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

$$\left(x-6\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$a\left(x-6\right)\left(x-1\right)$$

Evaluate

$$a\left(x-6\right)\left(x-1\right)$$