Solve by Factoring square root of t+8=2t+1

$\sqrt{t + 8} = 2 t + 1$

Step 1

Move all the expressions to the left side of the equation.

Subtract $2 t$ from both sides of the equation.

$\sqrt{t + 8} - 2 t = 1$

Subtract $1$ from both sides of the equation.

$\sqrt{t + 8} - 2 t - 1 = 0$

Step 2

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

Add $2 t$ to both sides of the equation.

$\sqrt{t + 8} - 1 = 2 t$

Add $1$ to both sides of the equation.

$\sqrt{t + 8} = 2 t + 1$

Step 3

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

${\sqrt{t + 8}}^{2} = {(2 t + 1)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 8}$ as ${(t + 8)}^{\frac{1}{2}}$ .

${({(t + 8)}^{\frac{1}{2}})}^{2} = {(2 t + 1)}^{2}$

Simplify the left side.

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

Multiply the exponents in ${({(t + 8)}^{\frac{1}{2}})}^{2}$ .

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

${(t + 8)}^{\frac{1}{2} \cdot 2} = {(2 t + 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(t + 8)}^{\frac{1}{2} \cdot 2} = {(2 t + 1)}^{2}$

Rewrite the expression.

${(t + 8)}^{1} = {(2 t + 1)}^{2}$

Simplify.

$t + 8 = {(2 t + 1)}^{2}$

Simplify the right side.

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

Rewrite ${(2 t + 1)}^{2}$ as $(2 t + 1) (2 t + 1)$ .

$t + 8 = (2 t + 1) (2 t + 1)$

Expand $(2 t + 1) (2 t + 1)$ using the FOIL Method.

Apply the distributive property.

$t + 8 = 2 t (2 t + 1) + 1 (2 t + 1)$

Apply the distributive property.

$t + 8 = 2 t (2 t) + 2 t \cdot 1 + 1 (2 t + 1)$

Apply the distributive property.

$t + 8 = 2 t (2 t) + 2 t \cdot 1 + 1 (2 t) + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$t + 8 = 2 \cdot 2 t \cdot t + 2 t \cdot 1 + 1 (2 t) + 1 \cdot 1$

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

Move $t$ .

$t + 8 = 2 \cdot 2 (t \cdot t) + 2 t \cdot 1 + 1 (2 t) + 1 \cdot 1$

Multiply $t$ by $t$ .

$t + 8 = 2 \cdot 2 t^{2} + 2 t \cdot 1 + 1 (2 t) + 1 \cdot 1$

Multiply $2$ by $2$ .

$t + 8 = 4 t^{2} + 2 t \cdot 1 + 1 (2 t) + 1 \cdot 1$

Multiply $2$ by $1$ .

$t + 8 = 4 t^{2} + 2 t + 1 (2 t) + 1 \cdot 1$

Multiply $2 t$ by $1$ .

$t + 8 = 4 t^{2} + 2 t + 2 t + 1 \cdot 1$

Multiply $1$ by $1$ .

$t + 8 = 4 t^{2} + 2 t + 2 t + 1$

Add $2 t$ and $2 t$ .

$t + 8 = 4 t^{2} + 4 t + 1$

Step 5

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.

$4 t^{2} + 4 t + 1 = t + 8$

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

Subtract $t$ from both sides of the equation.

$4 t^{2} + 4 t + 1 - t = 8$

Subtract $t$ from $4 t$ .

$4 t^{2} + 3 t + 1 = 8$

Subtract $8$ from both sides of the equation.

$4 t^{2} + 3 t + 1 - 8 = 0$

Subtract $8$ from $1$ .

$4 t^{2} + 3 t - 7 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 7 = - 28$ and whose sum is $b = 3$ .

Factor $3$ out of $3 t$ .

$4 t^{2} + 3 (t) - 7 = 0$

Rewrite $3$ as $- 4$ plus $7$

$4 t^{2} + (- 4 + 7) t - 7 = 0$

Apply the distributive property.

$4 t^{2} - 4 t + 7 t - 7 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 t^{2} - 4 t) + 7 t - 7 = 0$

Factor out the greatest common factor (GCF) from each group.

$4 t (t - 1) + 7 (t - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $t - 1$ .

$(t - 1) (4 t + 7) = 0$

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

$t - 1 = 0$

$4 t + 7 = 0$

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.

$t = 1$

Set $4 t + 7$ equal to $0$ and solve for $t$ .

Set $4 t + 7$ equal to $0$ .

$4 t + 7 = 0$

Solve $4 t + 7 = 0$ for $t$ .

Subtract $7$ from both sides of the equation.

$4 t = - 7$

Divide each term in $4 t = - 7$ by $4$ and simplify.

Divide each term in $4 t = - 7$ by $4$ .

$\frac{4 t}{4} = \frac{- 7}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 t}{4} = \frac{- 7}{4}$

Divide $t$ by $1$ .

$t = \frac{- 7}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$t = - \frac{7}{4}$

The final solution is all the values that make $(t - 1) (4 t + 7) = 0$ true.

$t = 1, - \frac{7}{4}$

Step 6

Exclude the solutions that do not make $\sqrt{t + 8} - 2 t - 1 = 0$ true.

$t = 1$

Solve by Factoring square root of t+8=2t+1

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