Solve 36x^2-60x+25-x^2

$$36 { x }^{ 2 } -60x+25- { x }^{ 2 }$$

Factor

$$5\left(x-1\right)\left(7x-5\right)$$

Solution Steps

Multiply and combine like terms.

$$35x^{2}-60x+25$$

Factor out 5.

$$5\left(7x^{2}-12x+5\right)$$

Consider 7x^{2}-12x+5. Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-12 ab=7\times 5=35$$

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 35.

$$-1,-35 -5,-7$$

Calculate the sum for each pair.

$$-1-35=-36 -5-7=-12$$

The solution is the pair that gives sum -12.

$$a=-7 b=-5$$

Rewrite 7x^{2}-12x+5 as \left(7x^{2}-7x\right)+\left(-5x+5\right).

$$\left(7x^{2}-7x\right)+\left(-5x+5\right)$$

Factor out 7x in the first and -5 in the second group.

$$7x\left(x-1\right)-5\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(7x-5\right)$$

Rewrite the complete factored expression.

$$5\left(x-1\right)\left(7x-5\right)$$

Evaluate

$$5\left(x-1\right)\left(7x-5\right)$$

Solution Steps

Combine 36x^{2} and -x^{2} to get 35x^{2}.

$$35x^{2}-60x+25$$