Solve 7x^2+52x-32

$$7 { x }^{ 2 } +52x-32$$

Factor

$$\left(7x-4\right)\left(x+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-32. To find a and b, set up a system to be solved.

$$a+b=52 ab=7\left(-32\right)=-224$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -224.

$$-1,224 -2,112 -4,56 -7,32 -8,28 -14,16$$

Calculate the sum for each pair.

$$-1+224=223 -2+112=110 -4+56=52 -7+32=25 -8+28=20 -14+16=2$$

The solution is the pair that gives sum 52.

$$a=-4 b=56$$

Rewrite 7x^{2}+52x-32 as \left(7x^{2}-4x\right)+\left(56x-32\right).

$$\left(7x^{2}-4x\right)+\left(56x-32\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(7x-4\right)+8\left(7x-4\right)$$

Factor out common term 7x-4 by using distributive property.

$$\left(7x-4\right)\left(x+8\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$7x^{2}+52x-32=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-52±\sqrt{52^{2}-4\times 7\left(-32\right)}}{2\times 7}$$

Square 52.

$$x=\frac{-52±\sqrt{2704-4\times 7\left(-32\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-52±\sqrt{2704-28\left(-32\right)}}{2\times 7}$$

Multiply -28 times -32.

$$x=\frac{-52±\sqrt{2704+896}}{2\times 7}$$

Add 2704 to 896.

$$x=\frac{-52±\sqrt{3600}}{2\times 7}$$

Take the square root of 3600.

$$x=\frac{-52±60}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-52±60}{14}$$

Now solve the equation x=\frac{-52±60}{14} when ± is plus. Add -52 to 60.

$$x=\frac{8}{14}$$

Reduce the fraction \frac{8}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{7}$$

Now solve the equation x=\frac{-52±60}{14} when ± is minus. Subtract 60 from -52.

$$x=-\frac{112}{14}$$

Divide -112 by 14.

$$x=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{4}{7} for x_{1} and -8 for x_{2}.

$$7x^{2}+52x-32=7\left(x-\frac{4}{7}\right)\left(x-\left(-8\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$7x^{2}+52x-32=7\left(x-\frac{4}{7}\right)\left(x+8\right)$$

Subtract \frac{4}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}+52x-32=7\times \frac{7x-4}{7}\left(x+8\right)$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7x^{2}+52x-32=\left(7x-4\right)\left(x+8\right)$$

Evaluate

$$\left(7x-4\right)\left(x+8\right)$$