Solve (x+2)(x+4)=51

$$(x+2)(x+4)=51$$

Solve for x

$$x=2\sqrt{13}-3\approx 4.211102551$$

$$x=-2\sqrt{13}-3\approx -10.211102551$$

Steps Using the Quadratic Formula

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

$$x^{2}+6x+8=51$$

Subtract 51 from both sides.

$$x^{2}+6x+8-51=0$$

Subtract 51 from 8 to get -43.

$$x^{2}+6x-43=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -43 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-43\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-43\right)}}{2}$$

Multiply -4 times -43.

$$x=\frac{-6±\sqrt{36+172}}{2}$$

Add 36 to 172.

$$x=\frac{-6±\sqrt{208}}{2}$$

Take the square root of 208.

$$x=\frac{-6±4\sqrt{13}}{2}$$

Now solve the equation x=\frac{-6±4\sqrt{13}}{2} when ± is plus. Add -6 to 4\sqrt{13}.

$$x=\frac{4\sqrt{13}-6}{2}$$

Divide -6+4\sqrt{13} by 2.

$$x=2\sqrt{13}-3$$

Now solve the equation x=\frac{-6±4\sqrt{13}}{2} when ± is minus. Subtract 4\sqrt{13} from -6.

$$x=\frac{-4\sqrt{13}-6}{2}$$

Divide -6-4\sqrt{13} by 2.

$$x=-2\sqrt{13}-3$$

The equation is now solved.

$$x=2\sqrt{13}-3 x=-2\sqrt{13}-3$$

Steps for Completing the Square

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

$$x^{2}+6x+8=51$$

Subtract 8 from both sides.

$$x^{2}+6x=51-8$$

Subtract 8 from 51 to get 43.

$$x^{2}+6x=43$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=43+3^{2}$$

Square 3.

$$x^{2}+6x+9=43+9$$

Add 43 to 9.

$$x^{2}+6x+9=52$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=52$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{52}$$

Simplify.

$$x+3=2\sqrt{13} x+3=-2\sqrt{13}$$

Subtract 3 from both sides of the equation.

$$x=2\sqrt{13}-3 x=-2\sqrt{13}-3$$