What is (v+sqrt(v^2+1))^2 ?

The solution to (v+sqrt(v^2+1))^2 is 2v^2+2vsqrt(v^2+1)+1

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simplify (v+sqrt(v^2+1))^2

simplify

(v + v^{2} + 1)^{2}

2 v^{2} + 2 v v^{2} + 1 + 1

Solution steps

(v + v^{2} + 1)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(v + v^{2} + 1)^{2} = v^{2} + 2 v v^{2} + 1 + (v^{2} + 1)^{2}

= v^{2} + 2 v v^{2} + 1 + (v^{2} + 1)^{2}

Simplify

v^{2} + 2 v v^{2} + 1 + (v^{2} + 1)^{2} : 2 v^{2} + 2 v v^{2} + 1 + 1

= 2 v^{2} + 2 v v^{2} + 1 + 1

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What is (v+sqrt(v^2+1))^2 ?
The solution to (v+sqrt(v^2+1))^2 is 2v^2+2vsqrt(v^2+1)+1