Solve xx+6x=6

$$xx+6x=6$$

Solve for x (complex solution)

$$x=\sqrt{15}-3\approx 0.872983346$$

$$x=-\left(\sqrt{15}+3\right)\approx -6.872983346$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

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

Subtract 6 from both sides.

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

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

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

Square 6.

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

Multiply -4 times -6.

$$x=\frac{-6±\sqrt{36+24}}{2}$$

Add 36 to 24.

$$x=\frac{-6±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{-6±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{-6±2\sqrt{15}}{2} when ± is plus. Add -6 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-6}{2}$$

Divide -6+2\sqrt{15} by 2.

$$x=\sqrt{15}-3$$

Now solve the equation x=\frac{-6±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from -6.

$$x=\frac{-2\sqrt{15}-6}{2}$$

Divide -6-2\sqrt{15} by 2.

$$x=-\sqrt{15}-3$$

The equation is now solved.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

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

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.

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

Square 3.

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

Add 6 to 9.

$$x^{2}+6x+9=15$$

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}.

$$\left(x+3\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{15}$$

Simplify.

$$x+3=\sqrt{15} x+3=-\sqrt{15}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$

Solve for x

$$x=\sqrt{15}-3\approx 0.872983346$$

$$x=-\sqrt{15}-3\approx -6.872983346$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

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

Subtract 6 from both sides.

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

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

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

Square 6.

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

Multiply -4 times -6.

$$x=\frac{-6±\sqrt{36+24}}{2}$$

Add 36 to 24.

$$x=\frac{-6±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{-6±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{-6±2\sqrt{15}}{2} when ± is plus. Add -6 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-6}{2}$$

Divide -6+2\sqrt{15} by 2.

$$x=\sqrt{15}-3$$

Now solve the equation x=\frac{-6±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from -6.

$$x=\frac{-2\sqrt{15}-6}{2}$$

Divide -6-2\sqrt{15} by 2.

$$x=-\sqrt{15}-3$$

The equation is now solved.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

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

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.

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

Square 3.

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

Add 6 to 9.

$$x^{2}+6x+9=15$$

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}.

$$\left(x+3\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{15}$$

Simplify.

$$x+3=\sqrt{15} x+3=-\sqrt{15}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$