Solve for t square root of 3t+1+ square root of 5-t=4

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

Step 1

Subtract $\sqrt{5 - t}$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 t + 1}$ as ${(3 t + 1)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(3 t + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 t + 1)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(4 - \sqrt{5 - t})}^{2}$ .

Rewrite ${(4 - \sqrt{5 - t})}^{2}$ as $(4 - \sqrt{5 - t}) (4 - \sqrt{5 - t})$ .

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

Expand $(4 - \sqrt{5 - t}) (4 - \sqrt{5 - t})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$3 t + 1 = 16 + 4 (- \sqrt{5 - t}) - \sqrt{5 - t} \cdot 4 - \sqrt{5 - t} (- \sqrt{5 - t})$

Multiply $- 1$ by $4$ .

$3 t + 1 = 16 - 4 \sqrt{5 - t} - \sqrt{5 - t} \cdot 4 - \sqrt{5 - t} (- \sqrt{5 - t})$

Multiply $4$ by $- 1$ .

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} - \sqrt{5 - t} (- \sqrt{5 - t})$

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

Multiply $- 1$ by $- 1$ .

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + 1 \sqrt{5 - t} \sqrt{5 - t}$

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

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + \sqrt{5 - t} \sqrt{5 - t}$

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

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {\sqrt{5 - t}}^{1} \sqrt{5 - t}$

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

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {\sqrt{5 - t}}^{1} {\sqrt{5 - t}}^{1}$

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

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {\sqrt{5 - t}}^{1 + 1}$

Add $1$ and $1$ .

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {\sqrt{5 - t}}^{2}$

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

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

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {({(5 - t)}^{\frac{1}{2}})}^{2}$

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

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {(5 - t)}^{\frac{1}{2} \cdot 2}$

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

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {(5 - t)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {(5 - t)}^{\frac{2}{2}}$

Rewrite the expression.

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + {(5 - t)}^{1}$

Simplify.

$3 t + 1 = 16 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} + 5 - t$

Add $16$ and $5$ .

$3 t + 1 = 21 - 4 \sqrt{5 - t} - 4 \sqrt{5 - t} - t$

Subtract $4 \sqrt{5 - t}$ from $- 4 \sqrt{5 - t}$ .

$3 t + 1 = 21 - 8 \sqrt{5 - t} - t$

Step 4

Solve for $- 8 \sqrt{5 - t}$ .

Rewrite the equation as $21 - 8 \sqrt{5 - t} - t = 3 t + 1$ .

$21 - 8 \sqrt{5 - t} - t = 3 t + 1$

Move all terms not containing $- 8 \sqrt{5 - t}$ to the right side of the equation.

Subtract $21$ from both sides of the equation.

$- 8 \sqrt{5 - t} - t = 3 t + 1 - 21$

Add $t$ to both sides of the equation.

$- 8 \sqrt{5 - t} = 3 t + 1 - 21 + t$

Add $3 t$ and $t$ .

$- 8 \sqrt{5 - t} = 4 t + 1 - 21$

Subtract $21$ from $1$ .

$- 8 \sqrt{5 - t} = 4 t - 20$

Step 5

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

${(- 8 \sqrt{5 - t})}^{2} = {(4 t - 20)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 8 {(5 - t)}^{\frac{1}{2}})}^{2} = {(4 t - 20)}^{2}$

Simplify the left side.

Simplify ${(- 8 {(5 - t)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 8 {(5 - t)}^{\frac{1}{2}}$ .

${(- 8)}^{2} {({(5 - t)}^{\frac{1}{2}})}^{2} = {(4 t - 20)}^{2}$

Raise $- 8$ to the power of $2$ .

$64 {({(5 - t)}^{\frac{1}{2}})}^{2} = {(4 t - 20)}^{2}$

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

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

$64 {(5 - t)}^{\frac{1}{2} \cdot 2} = {(4 t - 20)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$64 {(5 - t)}^{\frac{1}{2} \cdot 2} = {(4 t - 20)}^{2}$

Rewrite the expression.

$64 {(5 - t)}^{1} = {(4 t - 20)}^{2}$

Simplify.

$64 (5 - t) = {(4 t - 20)}^{2}$

Apply the distributive property.

$64 \cdot 5 + 64 (- t) = {(4 t - 20)}^{2}$

Multiply.

Multiply $64$ by $5$ .

$320 + 64 (- t) = {(4 t - 20)}^{2}$

Multiply $- 1$ by $64$ .

$320 - 64 t = {(4 t - 20)}^{2}$

Simplify the right side.

Simplify ${(4 t - 20)}^{2}$ .

Rewrite ${(4 t - 20)}^{2}$ as $(4 t - 20) (4 t - 20)$ .

$320 - 64 t = (4 t - 20) (4 t - 20)$

Expand $(4 t - 20) (4 t - 20)$ using the FOIL Method.

Apply the distributive property.

$320 - 64 t = 4 t (4 t - 20) - 20 (4 t - 20)$

Apply the distributive property.

$320 - 64 t = 4 t (4 t) + 4 t \cdot - 20 - 20 (4 t - 20)$

Apply the distributive property.

$320 - 64 t = 4 t (4 t) + 4 t \cdot - 20 - 20 (4 t) - 20 \cdot - 20$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$320 - 64 t = 4 \cdot 4 t \cdot t + 4 t \cdot - 20 - 20 (4 t) - 20 \cdot - 20$

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

Move $t$ .

$320 - 64 t = 4 \cdot 4 (t \cdot t) + 4 t \cdot - 20 - 20 (4 t) - 20 \cdot - 20$

Multiply $t$ by $t$ .

$320 - 64 t = 4 \cdot 4 t^{2} + 4 t \cdot - 20 - 20 (4 t) - 20 \cdot - 20$

Multiply $4$ by $4$ .

$320 - 64 t = 16 t^{2} + 4 t \cdot - 20 - 20 (4 t) - 20 \cdot - 20$

Multiply $- 20$ by $4$ .

$320 - 64 t = 16 t^{2} - 80 t - 20 (4 t) - 20 \cdot - 20$

Multiply $4$ by $- 20$ .

$320 - 64 t = 16 t^{2} - 80 t - 80 t - 20 \cdot - 20$

Multiply $- 20$ by $- 20$ .

$320 - 64 t = 16 t^{2} - 80 t - 80 t + 400$

Subtract $80 t$ from $- 80 t$ .

$320 - 64 t = 16 t^{2} - 160 t + 400$

Step 7

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.

$16 t^{2} - 160 t + 400 = 320 - 64 t$

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

Add $64 t$ to both sides of the equation.

$16 t^{2} - 160 t + 400 + 64 t = 320$

Add $- 160 t$ and $64 t$ .

$16 t^{2} - 96 t + 400 = 320$

Subtract $320$ from both sides of the equation.

$16 t^{2} - 96 t + 400 - 320 = 0$

Subtract $320$ from $400$ .

$16 t^{2} - 96 t + 80 = 0$

Factor the left side of the equation.

Factor $16$ out of $16 t^{2} - 96 t + 80$ .

Factor $16$ out of $16 t^{2}$ .

$16 (t^{2}) - 96 t + 80 = 0$

Factor $16$ out of $- 96 t$ .

$16 (t^{2}) + 16 (- 6 t) + 80 = 0$

Factor $16$ out of $80$ .

$16 t^{2} + 16 (- 6 t) + 16 \cdot 5 = 0$

Factor $16$ out of $16 t^{2} + 16 (- 6 t)$ .

$16 (t^{2} - 6 t) + 16 \cdot 5 = 0$

Factor $16$ out of $16 (t^{2} - 6 t) + 16 \cdot 5$ .

$16 (t^{2} - 6 t + 5) = 0$

Factor.

Factor $t^{2} - 6 t + 5$ 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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Write the factored form using these integers.

$16 ((t - 5) (t - 1)) = 0$

Remove unnecessary parentheses.

$16 (t - 5) (t - 1) = 0$

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

$t - 5 = 0$

$t - 1 = 0$

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 $t - 1$ equal to $0$ and solve for $t$ .

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

$t - 1 = 0$

Add $1$ to both sides of the equation.

$t = 1$

The final solution is all the values that make $16 (t - 5) (t - 1) = 0$ true.

$t = 5, 1$

Solve for t square root of 3t+1+ square root of 5-t=4

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