Solve (x+4)(x+2)=-7x-34

$$( x + 4 ) ( x + 2 ) = - 7 x - 34$$

Solve for x

$$x=-7$$

$$x=-6$$

Steps Using the Quadratic Formula

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

$$x^{2}+6x+8=-7x-34$$

Add 7x to both sides.

$$x^{2}+6x+8+7x=-34$$

Combine 6x and 7x to get 13x.

$$x^{2}+13x+8=-34$$

Add 34 to both sides.

$$x^{2}+13x+8+34=0$$

Add 8 and 34 to get 42.

$$x^{2}+13x+42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 13 for b, and 42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 42}}{2}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 42}}{2}$$

Multiply -4 times 42.

$$x=\frac{-13±\sqrt{169-168}}{2}$$

Add 169 to -168.

$$x=\frac{-13±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-13±1}{2}$$

Now solve the equation x=\frac{-13±1}{2} when ± is plus. Add -13 to 1.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

Now solve the equation x=\frac{-13±1}{2} when ± is minus. Subtract 1 from -13.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

The equation is now solved.

$$x=-6 x=-7$$

Steps for Completing the Square

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

$$x^{2}+6x+8=-7x-34$$

Add 7x to both sides.

$$x^{2}+6x+8+7x=-34$$

Combine 6x and 7x to get 13x.

$$x^{2}+13x+8=-34$$

Subtract 8 from both sides.

$$x^{2}+13x=-34-8$$

Subtract 8 from -34 to get -42.

$$x^{2}+13x=-42$$

Divide 13, the coefficient of the x term, by 2 to get \frac{13}{2}. Then add the square of \frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+13x+\left(\frac{13}{2}\right)^{2}=-42+\left(\frac{13}{2}\right)^{2}$$

Square \frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+13x+\frac{169}{4}=-42+\frac{169}{4}$$

Add -42 to \frac{169}{4}.

$$x^{2}+13x+\frac{169}{4}=\frac{1}{4}$$

Factor x^{2}+13x+\frac{169}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x+\frac{13}{2}=\frac{1}{2} x+\frac{13}{2}=-\frac{1}{2}$$

Subtract \frac{13}{2} from both sides of the equation.

$$x=-6 x=-7$$