Solve for t square root of 2t+15=t

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

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 t + 15}$ as ${(2 t + 15)}^{\frac{1}{2}}$ .

${({(2 t + 15)}^{\frac{1}{2}})}^{2} = t^{2}$

Simplify the left side.

Simplify ${({(2 t + 15)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 t + 15)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 t + 15)}^{\frac{1}{2} \cdot 2} = t^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 t + 15)}^{\frac{1}{2} \cdot 2} = t^{2}$

Rewrite the expression.

${(2 t + 15)}^{1} = t^{2}$

Simplify.

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

Step 3

Solve for $t$ .

Subtract $t^{2}$ from both sides of the equation.

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

Factor the left side of the equation.

Factor $- 1$ out of $2 t + 15 - t^{2}$ .

Reorder the expression.

Move $15$ .

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

Reorder $2 t$ and $- t^{2}$ .

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

Factor $- 1$ out of $- t^{2}$ .

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

Factor $- 1$ out of $2 t$ .

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

Rewrite $15$ as $- 1 (- 15)$ .

$- (t^{2}) - (- 2 t) - 1 \cdot - 15 = 0$

Factor $- 1$ out of $- (t^{2}) - (- 2 t)$ .

$- (t^{2} - 2 t) - 1 \cdot - 15 = 0$

Factor $- 1$ out of $- (t^{2} - 2 t) - 1 (- 15)$ .

$- (t^{2} - 2 t - 15) = 0$

Factor.

Factor $t^{2} - 2 t - 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$- ((t - 5) (t + 3)) = 0$

Remove unnecessary parentheses.

$- (t - 5) (t + 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$t - 5 = 0$

$t + 3 = 0$

Set $t - 5$ equal to $0$ and solve for $t$ .

Set $t - 5$ equal to $0$ .

$t - 5 = 0$

Add $5$ to both sides of the equation.

$t = 5$

Set $t + 3$ equal to $0$ and solve for $t$ .

Set $t + 3$ equal to $0$ .

$t + 3 = 0$

Subtract $3$ from both sides of the equation.

$t = - 3$

The final solution is all the values that make $- (t - 5) (t + 3) = 0$ true.

$t = 5, - 3$

Step 4

Exclude the solutions that do not make $\sqrt{2 t + 15} = t$ true.

$t = 5$

Solve for t square root of 2t+15=t

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