Solve 1+sqrt{v}=sqrt{v+7}

$$1 + \sqrt { v } = \sqrt { v + 7 }$$

Solve for v

$$v=9$$

Solution Steps

Square both sides of the equation.

$$\left(1+\sqrt{v}\right)^{2}=\left(\sqrt{v+7}\right)^{2}$$

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

$$1+2\sqrt{v}+\left(\sqrt{v}\right)^{2}=\left(\sqrt{v+7}\right)^{2}$$

Calculate \sqrt{v} to the power of 2 and get v.

$$1+2\sqrt{v}+v=\left(\sqrt{v+7}\right)^{2}$$

Calculate \sqrt{v+7} to the power of 2 and get v+7.

$$1+2\sqrt{v}+v=v+7$$

Subtract v from both sides.

$$1+2\sqrt{v}+v-v=7$$

Combine v and -v to get 0.

$$1+2\sqrt{v}=7$$

Subtract 1 from both sides.

$$2\sqrt{v}=7-1$$

Subtract 1 from 7 to get 6.

$$2\sqrt{v}=6$$

Divide both sides by 2.

$$\sqrt{v}=\frac{6}{2}$$

Divide 6 by 2 to get 3.

$$\sqrt{v}=3$$

Square both sides of the equation.

$$v=9$$

Substitute 9 for v in the equation 1+\sqrt{v}=\sqrt{v+7}.

$$1+\sqrt{9}=\sqrt{9+7}$$

Simplify. The value v=9 satisfies the equation.

$$4=4$$

Equation \sqrt{v}+1=\sqrt{v+7} has a unique solution.

$$v=9$$