Solve x^2+8x+7=2x+11

$$x ^ { 2 } + 8 x + 7 = 2 x + 11$$

Solve for x (complex solution)

$$x=\sqrt{13}-3\approx 0.605551275$$

$$x=-\left(\sqrt{13}+3\right)\approx -6.605551275$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$x^{2}+8x+7-2x=11$$

Combine 8x and -2x to get 6x.

$$x^{2}+6x+7=11$$

Subtract 11 from both sides.

$$x^{2}+6x+7-11=0$$

Subtract 11 from 7 to get -4.

$$x^{2}+6x-4=0$$

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

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

Square 6.

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

Multiply -4 times -4.

$$x=\frac{-6±\sqrt{36+16}}{2}$$

Add 36 to 16.

$$x=\frac{-6±\sqrt{52}}{2}$$

Take the square root of 52.

$$x=\frac{-6±2\sqrt{13}}{2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract 2x from both sides.

$$x^{2}+8x+7-2x=11$$

Combine 8x and -2x to get 6x.

$$x^{2}+6x+7=11$$

Subtract 7 from both sides.

$$x^{2}+6x=11-7$$

Subtract 7 from 11 to get 4.

$$x^{2}+6x=4$$

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.

$$x^{2}+6x+3^{2}=4+3^{2}$$

Square 3.

$$x^{2}+6x+9=4+9$$

Add 4 to 9.

$$x^{2}+6x+9=13$$

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}.

$$\left(x+3\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{13}$$

Simplify.

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

Subtract 3 from both sides of the equation.

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

Solve for x

$$x=\sqrt{13}-3\approx 0.605551275$$

$$x=-\sqrt{13}-3\approx -6.605551275$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$x^{2}+8x+7-2x=11$$

Combine 8x and -2x to get 6x.

$$x^{2}+6x+7=11$$

Subtract 11 from both sides.

$$x^{2}+6x+7-11=0$$

Subtract 11 from 7 to get -4.

$$x^{2}+6x-4=0$$

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

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

Square 6.

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

Multiply -4 times -4.

$$x=\frac{-6±\sqrt{36+16}}{2}$$

Add 36 to 16.

$$x=\frac{-6±\sqrt{52}}{2}$$

Take the square root of 52.

$$x=\frac{-6±2\sqrt{13}}{2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract 2x from both sides.

$$x^{2}+8x+7-2x=11$$

Combine 8x and -2x to get 6x.

$$x^{2}+6x+7=11$$

Subtract 7 from both sides.

$$x^{2}+6x=11-7$$

Subtract 7 from 11 to get 4.

$$x^{2}+6x=4$$

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.

$$x^{2}+6x+3^{2}=4+3^{2}$$

Square 3.

$$x^{2}+6x+9=4+9$$

Add 4 to 9.

$$x^{2}+6x+9=13$$

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}.

$$\left(x+3\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{13}$$

Simplify.

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

Subtract 3 from both sides of the equation.

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