Solve (a-4)(a+2)=7

$$( a - 4 ) ( a + 2 ) = 7$$

Solve for a

$$a=-3$$

$$a=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply a-4 by a+2 and combine like terms.

$$a^{2}-2a-8=7$$

Subtract 7 from both sides.

$$a^{2}-2a-8-7=0$$

Subtract 7 from -8 to get -15.

$$a^{2}-2a-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-15\right)}}{2}$$

Square -2.

$$a=\frac{-\left(-2\right)±\sqrt{4-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$a=\frac{-\left(-2\right)±\sqrt{4+60}}{2}$$

Add 4 to 60.

$$a=\frac{-\left(-2\right)±\sqrt{64}}{2}$$

Take the square root of 64.

$$a=\frac{-\left(-2\right)±8}{2}$$

The opposite of -2 is 2.

$$a=\frac{2±8}{2}$$

Now solve the equation a=\frac{2±8}{2} when ± is plus. Add 2 to 8.

$$a=\frac{10}{2}$$

Divide 10 by 2.

$$a=5$$

Now solve the equation a=\frac{2±8}{2} when ± is minus. Subtract 8 from 2.

$$a=-\frac{6}{2}$$

Divide -6 by 2.

$$a=-3$$

The equation is now solved.

$$a=5 a=-3$$

Steps for Completing the Square

Use the distributive property to multiply a-4 by a+2 and combine like terms.

$$a^{2}-2a-8=7$$

Add 8 to both sides.

$$a^{2}-2a=7+8$$

Add 7 and 8 to get 15.

$$a^{2}-2a=15$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-2a+1=15+1$$

Add 15 to 1.

$$a^{2}-2a+1=16$$

Factor a^{2}-2a+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-1\right)^{2}}=\sqrt{16}$$

Simplify.

$$a-1=4 a-1=-4$$

Add 1 to both sides of the equation.

$$a=5 a=-3$$