Solve (x-5)(x+7)=4

$$( x - 5 ) ( x + 7 ) = 4$$

Solve for x

$$x=2\sqrt{10}-1\approx 5.32455532$$

$$x=-2\sqrt{10}-1\approx -7.32455532$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-5 by x+7 and combine like terms.

$$x^{2}+2x-35=4$$

Subtract 4 from both sides.

$$x^{2}+2x-35-4=0$$

Subtract 4 from -35 to get -39.

$$x^{2}+2x-39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-39\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-39\right)}}{2}$$

Multiply -4 times -39.

$$x=\frac{-2±\sqrt{4+156}}{2}$$

Add 4 to 156.

$$x=\frac{-2±\sqrt{160}}{2}$$

Take the square root of 160.

$$x=\frac{-2±4\sqrt{10}}{2}$$

Now solve the equation x=\frac{-2±4\sqrt{10}}{2} when ± is plus. Add -2 to 4\sqrt{10}.

$$x=\frac{4\sqrt{10}-2}{2}$$

Divide -2+4\sqrt{10} by 2.

$$x=2\sqrt{10}-1$$

Now solve the equation x=\frac{-2±4\sqrt{10}}{2} when ± is minus. Subtract 4\sqrt{10} from -2.

$$x=\frac{-4\sqrt{10}-2}{2}$$

Divide -2-4\sqrt{10} by 2.

$$x=-2\sqrt{10}-1$$

The equation is now solved.

$$x=2\sqrt{10}-1 x=-2\sqrt{10}-1$$

Steps for Completing the Square

Use the distributive property to multiply x-5 by x+7 and combine like terms.

$$x^{2}+2x-35=4$$

Add 35 to both sides.

$$x^{2}+2x=4+35$$

Add 4 and 35 to get 39.

$$x^{2}+2x=39$$

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.

$$x^{2}+2x+1^{2}=39+1^{2}$$

Square 1.

$$x^{2}+2x+1=39+1$$

Add 39 to 1.

$$x^{2}+2x+1=40$$

Factor x^{2}+2x+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}.

$$\left(x+1\right)^{2}=40$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{40}$$

Simplify.

$$x+1=2\sqrt{10} x+1=-2\sqrt{10}$$

Subtract 1 from both sides of the equation.

$$x=2\sqrt{10}-1 x=-2\sqrt{10}-1$$