Solve (a-2)(a-2)=11

$$( a - 2 ) ( a - 2 ) = 11$$

Solve for a

$$a=\sqrt{11}+2\approx 5.31662479$$

$$a=2-\sqrt{11}\approx -1.31662479$$

Steps Using the Quadratic Formula

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

$$\left(a-2\right)^{2}=11$$

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

$$a^{2}-4a+4=11$$

Subtract 11 from both sides.

$$a^{2}-4a+4-11=0$$

Subtract 11 from 4 to get -7.

$$a^{2}-4a-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-7\right)}}{2}$$

Square -4.

$$a=\frac{-\left(-4\right)±\sqrt{16-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$a=\frac{-\left(-4\right)±\sqrt{16+28}}{2}$$

Add 16 to 28.

$$a=\frac{-\left(-4\right)±\sqrt{44}}{2}$$

Take the square root of 44.

$$a=\frac{-\left(-4\right)±2\sqrt{11}}{2}$$

The opposite of -4 is 4.

$$a=\frac{4±2\sqrt{11}}{2}$$

Now solve the equation a=\frac{4±2\sqrt{11}}{2} when ± is plus. Add 4 to 2\sqrt{11}.

$$a=\frac{2\sqrt{11}+4}{2}$$

Divide 4+2\sqrt{11} by 2.

$$a=\sqrt{11}+2$$

Now solve the equation a=\frac{4±2\sqrt{11}}{2} when ± is minus. Subtract 2\sqrt{11} from 4.

$$a=\frac{4-2\sqrt{11}}{2}$$

Divide 4-2\sqrt{11} by 2.

$$a=2-\sqrt{11}$$

The equation is now solved.

$$a=\sqrt{11}+2 a=2-\sqrt{11}$$

Steps for Completing the Square

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

$$\left(a-2\right)^{2}=11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2\right)^{2}}=\sqrt{11}$$

Simplify.

$$a-2=\sqrt{11} a-2=-\sqrt{11}$$

Add 2 to both sides of the equation.

$$a=\sqrt{11}+2 a=2-\sqrt{11}$$