Solve (x-12)(x-12)-6=384

$$(x-12)(x-12)-6=384$$

Solve for x

$$x=\sqrt{390}+12\approx 31.748417658$$

$$x=12-\sqrt{390}\approx -7.748417658$$

Steps Using the Quadratic Formula

Multiply x-12 and x-12 to get \left(x-12\right)^{2}.

$$\left(x-12\right)^{2}-6=384$$

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

$$x^{2}-24x+144-6=384$$

Subtract 6 from 144 to get 138.

$$x^{2}-24x+138=384$$

Subtract 384 from both sides.

$$x^{2}-24x+138-384=0$$

Subtract 384 from 138 to get -246.

$$x^{2}-24x-246=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and -246 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-246\right)}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-246\right)}}{2}$$

Multiply -4 times -246.

$$x=\frac{-\left(-24\right)±\sqrt{576+984}}{2}$$

Add 576 to 984.

$$x=\frac{-\left(-24\right)±\sqrt{1560}}{2}$$

Take the square root of 1560.

$$x=\frac{-\left(-24\right)±2\sqrt{390}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±2\sqrt{390}}{2}$$

Now solve the equation x=\frac{24±2\sqrt{390}}{2} when ± is plus. Add 24 to 2\sqrt{390}.

$$x=\frac{2\sqrt{390}+24}{2}$$

Divide 24+2\sqrt{390} by 2.

$$x=\sqrt{390}+12$$

Now solve the equation x=\frac{24±2\sqrt{390}}{2} when ± is minus. Subtract 2\sqrt{390} from 24.

$$x=\frac{24-2\sqrt{390}}{2}$$

Divide 24-2\sqrt{390} by 2.

$$x=12-\sqrt{390}$$

The equation is now solved.

$$x=\sqrt{390}+12 x=12-\sqrt{390}$$

Steps for Completing the Square

Multiply x-12 and x-12 to get \left(x-12\right)^{2}.

$$\left(x-12\right)^{2}-6=384$$

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

$$x^{2}-24x+144-6=384$$

Subtract 6 from 144 to get 138.

$$x^{2}-24x+138=384$$

Subtract 138 from both sides.

$$x^{2}-24x=384-138$$

Subtract 138 from 384 to get 246.

$$x^{2}-24x=246$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=246+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=246+144$$

Add 246 to 144.

$$x^{2}-24x+144=390$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=390$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{390}$$

Simplify.

$$x-12=\sqrt{390} x-12=-\sqrt{390}$$

Add 12 to both sides of the equation.

$$x=\sqrt{390}+12 x=12-\sqrt{390}$$