Solve for t square root of 4t+5=3- square root of t+5

$\sqrt{4 t + 5} = 3 - \sqrt{t + 5}$

Step 1

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

${\sqrt{4 t + 5}}^{2} = {(3 - \sqrt{t + 5})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 t + 5}$ as ${(4 t + 5)}^{\frac{1}{2}}$ .

${({(4 t + 5)}^{\frac{1}{2}})}^{2} = {(3 - \sqrt{t + 5})}^{2}$

Simplify the left side.

Simplify ${({(4 t + 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 t + 5)}^{\frac{1}{2}})}^{2}$ .

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

${(4 t + 5)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{t + 5})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 t + 5)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{t + 5})}^{2}$

Rewrite the expression.

${(4 t + 5)}^{1} = {(3 - \sqrt{t + 5})}^{2}$

Simplify.

$4 t + 5 = {(3 - \sqrt{t + 5})}^{2}$

Simplify the right side.

Simplify ${(3 - \sqrt{t + 5})}^{2}$ .

Rewrite ${(3 - \sqrt{t + 5})}^{2}$ as $(3 - \sqrt{t + 5}) (3 - \sqrt{t + 5})$ .

$4 t + 5 = (3 - \sqrt{t + 5}) (3 - \sqrt{t + 5})$

Expand $(3 - \sqrt{t + 5}) (3 - \sqrt{t + 5})$ using the FOIL Method.

Apply the distributive property.

$4 t + 5 = 3 (3 - \sqrt{t + 5}) - \sqrt{t + 5} (3 - \sqrt{t + 5})$

Apply the distributive property.

$4 t + 5 = 3 \cdot 3 + 3 (- \sqrt{t + 5}) - \sqrt{t + 5} (3 - \sqrt{t + 5})$

Apply the distributive property.

$4 t + 5 = 3 \cdot 3 + 3 (- \sqrt{t + 5}) - \sqrt{t + 5} \cdot 3 - \sqrt{t + 5} (- \sqrt{t + 5})$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$4 t + 5 = 9 + 3 (- \sqrt{t + 5}) - \sqrt{t + 5} \cdot 3 - \sqrt{t + 5} (- \sqrt{t + 5})$

Multiply $- 1$ by $3$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - \sqrt{t + 5} \cdot 3 - \sqrt{t + 5} (- \sqrt{t + 5})$

Multiply $3$ by $- 1$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} - \sqrt{t + 5} (- \sqrt{t + 5})$

Multiply $- \sqrt{t + 5} (- \sqrt{t + 5})$ .

Multiply $- 1$ by $- 1$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + 1 \sqrt{t + 5} \sqrt{t + 5}$

Multiply $\sqrt{t + 5}$ by $1$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + \sqrt{t + 5} \sqrt{t + 5}$

Raise $\sqrt{t + 5}$ to the power of $1$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {\sqrt{t + 5}}^{1} \sqrt{t + 5}$

Raise $\sqrt{t + 5}$ to the power of $1$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {\sqrt{t + 5}}^{1} {\sqrt{t + 5}}^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {\sqrt{t + 5}}^{1 + 1}$

Add $1$ and $1$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {\sqrt{t + 5}}^{2}$

Rewrite ${\sqrt{t + 5}}^{2}$ as $t + 5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 5}$ as ${(t + 5)}^{\frac{1}{2}}$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {({(t + 5)}^{\frac{1}{2}})}^{2}$

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

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {(t + 5)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {(t + 5)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {(t + 5)}^{\frac{2}{2}}$

Rewrite the expression.

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + {(t + 5)}^{1}$

Simplify.

$4 t + 5 = 9 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + t + 5$

Add $9$ and $5$ .

$4 t + 5 = 14 - 3 \sqrt{t + 5} - 3 \sqrt{t + 5} + t$

Subtract $3 \sqrt{t + 5}$ from $- 3 \sqrt{t + 5}$ .

$4 t + 5 = 14 - 6 \sqrt{t + 5} + t$

Step 3

Solve for $- 6 \sqrt{t + 5}$ .

Rewrite the equation as $14 - 6 \sqrt{t + 5} + t = 4 t + 5$ .

$14 - 6 \sqrt{t + 5} + t = 4 t + 5$

Move all terms not containing $- 6 \sqrt{t + 5}$ to the right side of the equation.

Subtract $14$ from both sides of the equation.

$- 6 \sqrt{t + 5} + t = 4 t + 5 - 14$

Subtract $t$ from both sides of the equation.

$- 6 \sqrt{t + 5} = 4 t + 5 - 14 - t$

Subtract $t$ from $4 t$ .

$- 6 \sqrt{t + 5} = 3 t + 5 - 14$

Subtract $14$ from $5$ .

$- 6 \sqrt{t + 5} = 3 t - 9$

Step 4

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

${(- 6 \sqrt{t + 5})}^{2} = {(3 t - 9)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 5}$ as ${(t + 5)}^{\frac{1}{2}}$ .

${(- 6 {(t + 5)}^{\frac{1}{2}})}^{2} = {(3 t - 9)}^{2}$

Simplify the left side.

Simplify ${(- 6 {(t + 5)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 6 {(t + 5)}^{\frac{1}{2}}$ .

${(- 6)}^{2} {({(t + 5)}^{\frac{1}{2}})}^{2} = {(3 t - 9)}^{2}$

Raise $- 6$ to the power of $2$ .

$36 {({(t + 5)}^{\frac{1}{2}})}^{2} = {(3 t - 9)}^{2}$

Multiply the exponents in ${({(t + 5)}^{\frac{1}{2}})}^{2}$ .

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

$36 {(t + 5)}^{\frac{1}{2} \cdot 2} = {(3 t - 9)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(t + 5)}^{\frac{1}{2} \cdot 2} = {(3 t - 9)}^{2}$

Rewrite the expression.

$36 {(t + 5)}^{1} = {(3 t - 9)}^{2}$

Simplify.

$36 (t + 5) = {(3 t - 9)}^{2}$

Apply the distributive property.

$36 t + 36 \cdot 5 = {(3 t - 9)}^{2}$

Multiply $36$ by $5$ .

$36 t + 180 = {(3 t - 9)}^{2}$

Simplify the right side.

Simplify ${(3 t - 9)}^{2}$ .

Rewrite ${(3 t - 9)}^{2}$ as $(3 t - 9) (3 t - 9)$ .

$36 t + 180 = (3 t - 9) (3 t - 9)$

Expand $(3 t - 9) (3 t - 9)$ using the FOIL Method.

Apply the distributive property.

$36 t + 180 = 3 t (3 t - 9) - 9 (3 t - 9)$

Apply the distributive property.

$36 t + 180 = 3 t (3 t) + 3 t \cdot - 9 - 9 (3 t - 9)$

Apply the distributive property.

$36 t + 180 = 3 t (3 t) + 3 t \cdot - 9 - 9 (3 t) - 9 \cdot - 9$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$36 t + 180 = 3 \cdot 3 t \cdot t + 3 t \cdot - 9 - 9 (3 t) - 9 \cdot - 9$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$36 t + 180 = 3 \cdot 3 (t \cdot t) + 3 t \cdot - 9 - 9 (3 t) - 9 \cdot - 9$

Multiply $t$ by $t$ .

$36 t + 180 = 3 \cdot 3 t^{2} + 3 t \cdot - 9 - 9 (3 t) - 9 \cdot - 9$

Multiply $3$ by $3$ .

$36 t + 180 = 9 t^{2} + 3 t \cdot - 9 - 9 (3 t) - 9 \cdot - 9$

Multiply $- 9$ by $3$ .

$36 t + 180 = 9 t^{2} - 27 t - 9 (3 t) - 9 \cdot - 9$

Multiply $3$ by $- 9$ .

$36 t + 180 = 9 t^{2} - 27 t - 27 t - 9 \cdot - 9$

Multiply $- 9$ by $- 9$ .

$36 t + 180 = 9 t^{2} - 27 t - 27 t + 81$

Subtract $27 t$ from $- 27 t$ .

$36 t + 180 = 9 t^{2} - 54 t + 81$

Step 6

Solve for $t$ .

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 t^{2} - 54 t + 81 = 36 t + 180$

Move all terms containing $t$ to the left side of the equation.

Subtract $36 t$ from both sides of the equation.

$9 t^{2} - 54 t + 81 - 36 t = 180$

Subtract $36 t$ from $- 54 t$ .

$9 t^{2} - 90 t + 81 = 180$

Subtract $180$ from both sides of the equation.

$9 t^{2} - 90 t + 81 - 180 = 0$

Subtract $180$ from $81$ .

$9 t^{2} - 90 t - 99 = 0$

Factor the left side of the equation.

Factor $9$ out of $9 t^{2} - 90 t - 99$ .

Factor $9$ out of $9 t^{2}$ .

$9 (t^{2}) - 90 t - 99 = 0$

Factor $9$ out of $- 90 t$ .

$9 (t^{2}) + 9 (- 10 t) - 99 = 0$

Factor $9$ out of $- 99$ .

$9 t^{2} + 9 (- 10 t) + 9 \cdot - 11 = 0$

Factor $9$ out of $9 t^{2} + 9 (- 10 t)$ .

$9 (t^{2} - 10 t) + 9 \cdot - 11 = 0$

Factor $9$ out of $9 (t^{2} - 10 t) + 9 \cdot - 11$ .

$9 (t^{2} - 10 t - 11) = 0$

Factor.

Factor $t^{2} - 10 t - 11$ 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 $- 11$ and whose sum is $- 10$ .

$- 11, 1$

Write the factored form using these integers.

$9 ((t - 11) (t + 1)) = 0$

Remove unnecessary parentheses.

$9 (t - 11) (t + 1) = 0$

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

$t - 11 = 0$

$t + 1 = 0$

Set $t - 11$ equal to $0$ and solve for $t$ .

Set $t - 11$ equal to $0$ .

$t - 11 = 0$

Add $11$ to both sides of the equation.

$t = 11$

Set $t + 1$ equal to $0$ and solve for $t$ .

Set $t + 1$ equal to $0$ .

$t + 1 = 0$

Subtract $1$ from both sides of the equation.

$t = - 1$

The final solution is all the values that make $9 (t - 11) (t + 1) = 0$ true.

$t = 11, - 1$

Step 7

Exclude the solutions that do not make $\sqrt{4 t + 5} = 3 - \sqrt{t + 5}$ true.

$t = - 1$

Solve for t square root of 4t+5=3- square root of t+5

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