Solve 3x^2=12-6x

$$3 x ^ { 2 } = 12 - 6 x$$

Solve for x (complex solution)

$$x=\sqrt{5}-1\approx 1.236067977$$

$$x=-\left(\sqrt{5}+1\right)\approx -3.236067977$$

Steps Using the Quadratic Formula

Subtract 12 from both sides.

$$3x^{2}-12=-6x$$

Add 6x to both sides.

$$3x^{2}-12+6x=0$$

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.

$$3x^{2}+6x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 6 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3\left(-12\right)}}{2\times 3}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 3\left(-12\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-6±\sqrt{36-12\left(-12\right)}}{2\times 3}$$

Multiply -12 times -12.

$$x=\frac{-6±\sqrt{36+144}}{2\times 3}$$

Add 36 to 144.

$$x=\frac{-6±\sqrt{180}}{2\times 3}$$

Take the square root of 180.

$$x=\frac{-6±6\sqrt{5}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-6±6\sqrt{5}}{6}$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{6} when ± is plus. Add -6 to 6\sqrt{5}.

$$x=\frac{6\sqrt{5}-6}{6}$$

Divide -6+6\sqrt{5} by 6.

$$x=\sqrt{5}-1$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{6} when ± is minus. Subtract 6\sqrt{5} from -6.

$$x=\frac{-6\sqrt{5}-6}{6}$$

Divide -6-6\sqrt{5} by 6.

$$x=-\sqrt{5}-1$$

The equation is now solved.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$

Steps for Completing the Square

Add 6x to both sides.

$$3x^{2}+6x=12$$

Divide both sides by 3.

$$\frac{3x^{2}+6x}{3}=\frac{12}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{6}{3}x=\frac{12}{3}$$

Divide 6 by 3.

$$x^{2}+2x=\frac{12}{3}$$

Divide 12 by 3.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=4+1^{2}$$

Square 1.

$$x^{2}+2x+1=4+1$$

Add 4 to 1.

$$x^{2}+2x+1=5$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{5}$$

Simplify.

$$x+1=\sqrt{5} x+1=-\sqrt{5}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$

Solve for x

$$x=\sqrt{5}-1\approx 1.236067977$$

$$x=-\sqrt{5}-1\approx -3.236067977$$

Steps Using the Quadratic Formula

Subtract 12 from both sides.

$$3x^{2}-12=-6x$$

Add 6x to both sides.

$$3x^{2}-12+6x=0$$

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.

$$3x^{2}+6x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 6 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3\left(-12\right)}}{2\times 3}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 3\left(-12\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-6±\sqrt{36-12\left(-12\right)}}{2\times 3}$$

Multiply -12 times -12.

$$x=\frac{-6±\sqrt{36+144}}{2\times 3}$$

Add 36 to 144.

$$x=\frac{-6±\sqrt{180}}{2\times 3}$$

Take the square root of 180.

$$x=\frac{-6±6\sqrt{5}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-6±6\sqrt{5}}{6}$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{6} when ± is plus. Add -6 to 6\sqrt{5}.

$$x=\frac{6\sqrt{5}-6}{6}$$

Divide -6+6\sqrt{5} by 6.

$$x=\sqrt{5}-1$$

Now solve the equation x=\frac{-6±6\sqrt{5}}{6} when ± is minus. Subtract 6\sqrt{5} from -6.

$$x=\frac{-6\sqrt{5}-6}{6}$$

Divide -6-6\sqrt{5} by 6.

$$x=-\sqrt{5}-1$$

The equation is now solved.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$

Steps for Completing the Square

Add 6x to both sides.

$$3x^{2}+6x=12$$

Divide both sides by 3.

$$\frac{3x^{2}+6x}{3}=\frac{12}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{6}{3}x=\frac{12}{3}$$

Divide 6 by 3.

$$x^{2}+2x=\frac{12}{3}$$

Divide 12 by 3.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=4+1^{2}$$

Square 1.

$$x^{2}+2x+1=4+1$$

Add 4 to 1.

$$x^{2}+2x+1=5$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{5}$$

Simplify.

$$x+1=\sqrt{5} x+1=-\sqrt{5}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{5}-1 x=-\sqrt{5}-1$$