Solve x^2+6x+9=2

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

Solve for x (complex solution)

$$x=\sqrt{2}-3\approx -1.585786438$$

$$x=-\left(\sqrt{2}+3\right)\approx -4.414213562$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

Subtract 2 from both sides of the equation.

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

Subtracting 2 from itself leaves 0.

$$x^{2}+6x+9-2=0$$

Subtract 2 from 9.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 7}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-6±\sqrt{36-28}}{2}$$

Add 36 to -28.

$$x=\frac{-6±\sqrt{8}}{2}$$

Take the square root of 8.

$$x=\frac{-6±2\sqrt{2}}{2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 3 from both sides of the equation.

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

Solve for x

$$x=\sqrt{2}-3\approx -1.585786438$$

$$x=-\sqrt{2}-3\approx -4.414213562$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

Subtract 2 from both sides of the equation.

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

Subtracting 2 from itself leaves 0.

$$x^{2}+6x+9-2=0$$

Subtract 2 from 9.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 7}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-6±\sqrt{36-28}}{2}$$

Add 36 to -28.

$$x=\frac{-6±\sqrt{8}}{2}$$

Take the square root of 8.

$$x=\frac{-6±2\sqrt{2}}{2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 3 from both sides of the equation.

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