Solve 54=(x-2)^2

$$54 = ( x - 2 ) ^ { 2 }$$

Solve for x

$$x=3\sqrt{6}+2\approx 9.348469228$$

$$x=2-3\sqrt{6}\approx -5.348469228$$

Steps Using the Quadratic Formula

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

$$54=x^{2}-4x+4$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-4x+4=54$$

Subtract 54 from both sides.

$$x^{2}-4x+4-54=0$$

Subtract 54 from 4 to get -50.

$$x^{2}-4x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-50\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-50\right)}}{2}$$

Multiply -4 times -50.

$$x=\frac{-\left(-4\right)±\sqrt{16+200}}{2}$$

Add 16 to 200.

$$x=\frac{-\left(-4\right)±\sqrt{216}}{2}$$

Take the square root of 216.

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

The opposite of -4 is 4.

$$x=\frac{4±6\sqrt{6}}{2}$$

Now solve the equation x=\frac{4±6\sqrt{6}}{2} when ± is plus. Add 4 to 6\sqrt{6}.

$$x=\frac{6\sqrt{6}+4}{2}$$

Divide 4+6\sqrt{6} by 2.

$$x=3\sqrt{6}+2$$

Now solve the equation x=\frac{4±6\sqrt{6}}{2} when ± is minus. Subtract 6\sqrt{6} from 4.

$$x=\frac{4-6\sqrt{6}}{2}$$

Divide 4-6\sqrt{6} by 2.

$$x=2-3\sqrt{6}$$

The equation is now solved.

$$x=3\sqrt{6}+2 x=2-3\sqrt{6}$$

Steps for Completing the Square

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

$$54=x^{2}-4x+4$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-4x+4=54$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=54$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{54}$$

Simplify.

$$x-2=3\sqrt{6} x-2=-3\sqrt{6}$$

Add 2 to both sides of the equation.

$$x=3\sqrt{6}+2 x=2-3\sqrt{6}$$