Solve sqrt{2t+15}=t

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

Solve for t

$$t=5$$

Solution Steps

Square both sides of the equation.

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

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

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

Subtract t^{2} from both sides.

$$2t+15-t^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-t^{2}+2t+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -t^{2}+at+bt+15. To find a and b, set up a system to be solved.

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

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. 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 -t^{2}+2t+15 as \left(-t^{2}+5t\right)+\left(-3t+15\right).

$$\left(-t^{2}+5t\right)+\left(-3t+15\right)$$

Factor out -t in the first and -3 in the second group.

$$-t\left(t-5\right)-3\left(t-5\right)$$

Factor out common term t-5 by using distributive property.

$$\left(t-5\right)\left(-t-3\right)$$

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

$$t=5 t=-3$$

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

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

Simplify. The value t=5 satisfies the equation.

$$5=5$$

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

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

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

$$3=-3$$

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

$$t=5$$