Solve 3(x-1)^2+12=0

$$3 ( x - 1 ) ^ { 2 } + 12 = 0$$

Solve for x (complex solution)

$$x=1-2i$$

$$x=1+2i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$3\left(x^{2}-2x+1\right)+12=0$$

Use the distributive property to multiply 3 by x^{2}-2x+1.

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

Add 3 and 12 to get 15.

$$3x^{2}-6x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\times 15}}{2\times 3}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3\times 15}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12\times 15}}{2\times 3}$$

Multiply -12 times 15.

$$x=\frac{-\left(-6\right)±\sqrt{36-180}}{2\times 3}$$

Add 36 to -180.

$$x=\frac{-\left(-6\right)±\sqrt{-144}}{2\times 3}$$

Take the square root of -144.

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

The opposite of -6 is 6.

$$x=\frac{6±12i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{6±12i}{6}$$

Now solve the equation x=\frac{6±12i}{6} when ± is plus. Add 6 to 12i.

$$x=\frac{6+12i}{6}$$

Divide 6+12i by 6.

$$x=1+2i$$

Now solve the equation x=\frac{6±12i}{6} when ± is minus. Subtract 12i from 6.

$$x=\frac{6-12i}{6}$$

Divide 6-12i by 6.

$$x=1-2i$$

The equation is now solved.

$$x=1+2i x=1-2i$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$3\left(x^{2}-2x+1\right)+12=0$$

Use the distributive property to multiply 3 by x^{2}-2x+1.

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

Add 3 and 12 to get 15.

$$3x^{2}-6x+15=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-6x=-15$$

Divide both sides by 3.

$$\frac{3x^{2}-6x}{3}=-\frac{15}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{6}{3}\right)x=-\frac{15}{3}$$

Divide -6 by 3.

$$x^{2}-2x=-\frac{15}{3}$$

Divide -15 by 3.

$$x^{2}-2x=-5$$

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.

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

Add -5 to 1.

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

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}.

$$\left(x-1\right)^{2}=-4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-4}$$

Simplify.

$$x-1=2i x-1=-2i$$

Add 1 to both sides of the equation.

$$x=1+2i x=1-2i$$