Solve {left(sqrt{23}-1right)}^2+2(sqrt{23}-1)+2

$${ \left( \sqrt{ 23 } -1 \right) }^{ 2 } +2( \sqrt{ 23 } -1)+2$$

Evaluate

$$24$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{23}-1\right)^{2}.

$$\left(\sqrt{23}\right)^{2}-2\sqrt{23}+1+2\left(\sqrt{23}-1\right)+2$$

The square of \sqrt{23} is 23.

$$23-2\sqrt{23}+1+2\left(\sqrt{23}-1\right)+2$$

Add 23 and 1 to get 24.

$$24-2\sqrt{23}+2\left(\sqrt{23}-1\right)+2$$

Use the distributive property to multiply 2 by \sqrt{23}-1.

$$24-2\sqrt{23}+2\sqrt{23}-2+2$$

Combine -2\sqrt{23} and 2\sqrt{23} to get 0.

$$24-2+2$$

Subtract 2 from 24 to get 22.

$$22+2$$

Add 22 and 2 to get 24.

$$24$$

Factor

$$2^{3}\times 3$$