Solve 2x(x+6)=12

$$2 x ( x + 6 ) = 12$$

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

Use the distributive property to multiply 2x by x+6.

$$2x^{2}+12x=12$$

Subtract 12 from both sides.

$$2x^{2}+12x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 12 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2\left(-12\right)}}{2\times 2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2\left(-12\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-12±\sqrt{144-8\left(-12\right)}}{2\times 2}$$

Multiply -8 times -12.

$$x=\frac{-12±\sqrt{144+96}}{2\times 2}$$

Add 144 to 96.

$$x=\frac{-12±\sqrt{240}}{2\times 2}$$

Take the square root of 240.

$$x=\frac{-12±4\sqrt{15}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-12±4\sqrt{15}}{4}$$

Now solve the equation x=\frac{-12±4\sqrt{15}}{4} when ± is plus. Add -12 to 4\sqrt{15}.

$$x=\frac{4\sqrt{15}-12}{4}$$

Divide -12+4\sqrt{15} by 4.

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

Now solve the equation x=\frac{-12±4\sqrt{15}}{4} when ± is minus. Subtract 4\sqrt{15} from -12.

$$x=\frac{-4\sqrt{15}-12}{4}$$

Divide -12-4\sqrt{15} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 2x by x+6.

$$2x^{2}+12x=12$$

Divide both sides by 2.

$$\frac{2x^{2}+12x}{2}=\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{12}{2}x=\frac{12}{2}$$

Divide 12 by 2.

$$x^{2}+6x=\frac{12}{2}$$

Divide 12 by 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

Use the distributive property to multiply 2x by x+6.

$$2x^{2}+12x=12$$

Subtract 12 from both sides.

$$2x^{2}+12x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 12 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 2\left(-12\right)}}{2\times 2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 2\left(-12\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-12±\sqrt{144-8\left(-12\right)}}{2\times 2}$$

Multiply -8 times -12.

$$x=\frac{-12±\sqrt{144+96}}{2\times 2}$$

Add 144 to 96.

$$x=\frac{-12±\sqrt{240}}{2\times 2}$$

Take the square root of 240.

$$x=\frac{-12±4\sqrt{15}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-12±4\sqrt{15}}{4}$$

Now solve the equation x=\frac{-12±4\sqrt{15}}{4} when ± is plus. Add -12 to 4\sqrt{15}.

$$x=\frac{4\sqrt{15}-12}{4}$$

Divide -12+4\sqrt{15} by 4.

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

Now solve the equation x=\frac{-12±4\sqrt{15}}{4} when ± is minus. Subtract 4\sqrt{15} from -12.

$$x=\frac{-4\sqrt{15}-12}{4}$$

Divide -12-4\sqrt{15} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 2x by x+6.

$$2x^{2}+12x=12$$

Divide both sides by 2.

$$\frac{2x^{2}+12x}{2}=\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{12}{2}x=\frac{12}{2}$$

Divide 12 by 2.

$$x^{2}+6x=\frac{12}{2}$$

Divide 12 by 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.

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