Solve a^2-32-4a

$$a ^ { 2 } - 32 - 4 a$$

Factor

$$\left(a-8\right)\left(a+4\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$a^{2}-4a-32$$

Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-32. To find p and q, set up a system to be solved.

$$p+q=-4 pq=1\left(-32\right)=-32$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -32.

$$1,-32 2,-16 4,-8$$

Calculate the sum for each pair.

$$1-32=-31 2-16=-14 4-8=-4$$

The solution is the pair that gives sum -4.

$$p=-8 q=4$$

Rewrite a^{2}-4a-32 as \left(a^{2}-8a\right)+\left(4a-32\right).

$$\left(a^{2}-8a\right)+\left(4a-32\right)$$

Factor out a in the first and 4 in the second group.

$$a\left(a-8\right)+4\left(a-8\right)$$

Factor out common term a-8 by using distributive property.

$$\left(a-8\right)\left(a+4\right)$$

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.

$$a^{2}-4a-32=0$$

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.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-32\right)}}{2}$$

Square -4.

$$a=\frac{-\left(-4\right)±\sqrt{16-4\left(-32\right)}}{2}$$

Multiply -4 times -32.

$$a=\frac{-\left(-4\right)±\sqrt{16+128}}{2}$$

Add 16 to 128.

$$a=\frac{-\left(-4\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$a=\frac{-\left(-4\right)±12}{2}$$

The opposite of -4 is 4.

$$a=\frac{4±12}{2}$$

Now solve the equation a=\frac{4±12}{2} when ± is plus. Add 4 to 12.

$$a=\frac{16}{2}$$

Divide 16 by 2.

$$a=8$$

Now solve the equation a=\frac{4±12}{2} when ± is minus. Subtract 12 from 4.

$$a=-\frac{8}{2}$$

Divide -8 by 2.

$$a=-4$$

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

$$a^{2}-4a-32=\left(a-8\right)\left(a-\left(-4\right)\right)$$

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

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

Evaluate

$$\left(a-8\right)\left(a+4\right)$$