Solve v=sqrt{2v+15}

$$v = \sqrt { 2 v + 15 }$$

Solve for v

$$v=5$$

Solution Steps

Square both sides of the equation.

$$v^{2}=\left(\sqrt{2v+15}\right)^{2}$$

Calculate \sqrt{2v+15} to the power of 2 and get 2v+15.

$$v^{2}=2v+15$$

Subtract 2v from both sides.

$$v^{2}-2v=15$$

Subtract 15 from both sides.

$$v^{2}-2v-15=0$$

To solve the equation, factor v^{2}-2v-15 using formula v^{2}+\left(a+b\right)v+ab=\left(v+a\right)\left(v+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -2.

$$a=-5 b=3$$

Rewrite factored expression \left(v+a\right)\left(v+b\right) using the obtained values.

$$\left(v-5\right)\left(v+3\right)$$

To find equation solutions, solve v-5=0 and v+3=0.

$$v=5 v=-3$$

Substitute 5 for v in the equation v=\sqrt{2v+15}.

$$5=\sqrt{2\times 5+15}$$

Simplify. The value v=5 satisfies the equation.

$$5=5$$

Substitute -3 for v in the equation v=\sqrt{2v+15}.

$$-3=\sqrt{2\left(-3\right)+15}$$

Simplify. The value v=-3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-3=3$$

Equation v=\sqrt{2v+15} has a unique solution.

$$v=5$$