Solve left(2x-12right)^2+4x^2=48

$${ \left(2x-12 \right) }^{ 2 } +4 { x }^{ 2 } =48$$

Solve for x (complex solution)

$$x=3+\sqrt{3}i\approx 3+1.732050808i$$

$$x=-\sqrt{3}i+3\approx 3-1.732050808i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-12\right)^{2}.

$$4x^{2}-48x+144+4x^{2}=48$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}-48x+144=48$$

Subtract 48 from both sides.

$$8x^{2}-48x+144-48=0$$

Subtract 48 from 144 to get 96.

$$8x^{2}-48x+96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -48 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 8\times 96}}{2\times 8}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 8\times 96}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-48\right)±\sqrt{2304-32\times 96}}{2\times 8}$$

Multiply -32 times 96.

$$x=\frac{-\left(-48\right)±\sqrt{2304-3072}}{2\times 8}$$

Add 2304 to -3072.

$$x=\frac{-\left(-48\right)±\sqrt{-768}}{2\times 8}$$

Take the square root of -768.

$$x=\frac{-\left(-48\right)±16\sqrt{3}i}{2\times 8}$$

The opposite of -48 is 48.

$$x=\frac{48±16\sqrt{3}i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{48±16\sqrt{3}i}{16}$$

Now solve the equation x=\frac{48±16\sqrt{3}i}{16} when ± is plus. Add 48 to 16i\sqrt{3}.

$$x=\frac{48+16\sqrt{3}i}{16}$$

Divide 48+16i\sqrt{3} by 16.

$$x=3+\sqrt{3}i$$

Now solve the equation x=\frac{48±16\sqrt{3}i}{16} when ± is minus. Subtract 16i\sqrt{3} from 48.

$$x=\frac{-16\sqrt{3}i+48}{16}$$

Divide 48-16i\sqrt{3} by 16.

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

The equation is now solved.

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

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-12\right)^{2}.

$$4x^{2}-48x+144+4x^{2}=48$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}-48x+144=48$$

Subtract 144 from both sides.

$$8x^{2}-48x=48-144$$

Subtract 144 from 48 to get -96.

$$8x^{2}-48x=-96$$

Divide both sides by 8.

$$\frac{8x^{2}-48x}{8}=-\frac{96}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{48}{8}\right)x=-\frac{96}{8}$$

Divide -48 by 8.

$$x^{2}-6x=-\frac{96}{8}$$

Divide -96 by 8.

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

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.

$$x^{2}-6x+\left(-3\right)^{2}=-12+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-12+9$$

Add -12 to 9.

$$x^{2}-6x+9=-3$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 3 to both sides of the equation.

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