Solve (12sqrt{14}+1)^2-(12sqrt{14}+1)+(1-12sqrt{14})

$$( 12 \sqrt { 14 } + 1 ) ^ { 2 } - ( 12 \sqrt { 14 } + 1 ) + ( 1 - 12 \sqrt { 14 } )$$

Evaluate

$$2017$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12\sqrt{14}+1\right)^{2}.

$$144\left(\sqrt{14}\right)^{2}+24\sqrt{14}+1-\left(12\sqrt{14}+1\right)+1-12\sqrt{14}$$

The square of \sqrt{14} is 14.

$$144\times 14+24\sqrt{14}+1-\left(12\sqrt{14}+1\right)+1-12\sqrt{14}$$

Multiply 144 and 14 to get 2016.

$$2016+24\sqrt{14}+1-\left(12\sqrt{14}+1\right)+1-12\sqrt{14}$$

Add 2016 and 1 to get 2017.

$$2017+24\sqrt{14}-\left(12\sqrt{14}+1\right)+1-12\sqrt{14}$$

To find the opposite of 12\sqrt{14}+1, find the opposite of each term.

$$2017+24\sqrt{14}-12\sqrt{14}-1+1-12\sqrt{14}$$

Combine 24\sqrt{14} and -12\sqrt{14} to get 12\sqrt{14}.

$$2017+12\sqrt{14}-1+1-12\sqrt{14}$$

Subtract 1 from 2017 to get 2016.

$$2016+12\sqrt{14}+1-12\sqrt{14}$$

Add 2016 and 1 to get 2017.

$$2017+12\sqrt{14}-12\sqrt{14}$$

Combine 12\sqrt{14} and -12\sqrt{14} to get 0.

$$2017$$

Factor

$$2017$$