Solve for x square root of 3x-8+1 = square root of x+5

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

Subtract $1$ from both sides of the equation.

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

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

${\sqrt{3 x - 8}}^{2} = {(\sqrt{x + 5} - 1)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 8}$ as ${(3 x - 8)}^{\frac{1}{2}}$ .

${({(3 x - 8)}^{\frac{1}{2}})}^{2} = {(\sqrt{x + 5} - 1)}^{2}$

Simplify the left side.

Simplify ${({(3 x - 8)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x - 8)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x - 8)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 5} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x - 8)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 5} - 1)}^{2}$

Rewrite the expression.

${(3 x - 8)}^{1} = {(\sqrt{x + 5} - 1)}^{2}$

Simplify.

$3 x - 8 = {(\sqrt{x + 5} - 1)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 5} - 1)}^{2}$ .

Rewrite ${(\sqrt{x + 5} - 1)}^{2}$ as $(\sqrt{x + 5} - 1) (\sqrt{x + 5} - 1)$ .

$3 x - 8 = (\sqrt{x + 5} - 1) (\sqrt{x + 5} - 1)$

Expand $(\sqrt{x + 5} - 1) (\sqrt{x + 5} - 1)$ using the FOIL Method.

Apply the distributive property.

$3 x - 8 = \sqrt{x + 5} (\sqrt{x + 5} - 1) - 1 (\sqrt{x + 5} - 1)$

Apply the distributive property.

$3 x - 8 = \sqrt{x + 5} \sqrt{x + 5} + \sqrt{x + 5} \cdot - 1 - 1 (\sqrt{x + 5} - 1)$

Apply the distributive property.

$3 x - 8 = \sqrt{x + 5} \sqrt{x + 5} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 5} \sqrt{x + 5}$ .

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

$3 x - 8 = {\sqrt{x + 5}}^{1} \sqrt{x + 5} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

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

$3 x - 8 = {\sqrt{x + 5}}^{1} {\sqrt{x + 5}}^{1} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

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

$3 x - 8 = {\sqrt{x + 5}}^{1 + 1} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

Add $1$ and $1$ .

$3 x - 8 = {\sqrt{x + 5}}^{2} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

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

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

$3 x - 8 = {({(x + 5)}^{\frac{1}{2}})}^{2} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

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

$3 x - 8 = {(x + 5)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

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

$3 x - 8 = {(x + 5)}^{\frac{2}{2}} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 x - 8 = {(x + 5)}^{\frac{2}{2}} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

Rewrite the expression.

$3 x - 8 = {(x + 5)}^{1} + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

Simplify.

$3 x - 8 = x + 5 + \sqrt{x + 5} \cdot - 1 - 1 \sqrt{x + 5} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{x + 5}$ .

$3 x - 8 = x + 5 - 1 \cdot \sqrt{x + 5} - 1 \sqrt{x + 5} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{x + 5}$ as $- \sqrt{x + 5}$ .

$3 x - 8 = x + 5 - \sqrt{x + 5} - 1 \sqrt{x + 5} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{x + 5}$ as $- \sqrt{x + 5}$ .

$3 x - 8 = x + 5 - \sqrt{x + 5} - \sqrt{x + 5} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

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

Add $5$ and $1$ .

$3 x - 8 = x + 6 - \sqrt{x + 5} - \sqrt{x + 5}$

Subtract $\sqrt{x + 5}$ from $- \sqrt{x + 5}$ .

$3 x - 8 = x + 6 - 2 \sqrt{x + 5}$

Solve for $- 2 \sqrt{x + 5}$ .

Rewrite the equation as $x + 6 - 2 \sqrt{x + 5} = 3 x - 8$ .

$x + 6 - 2 \sqrt{x + 5} = 3 x - 8$

Move all terms not containing $- 2 \sqrt{x + 5}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$6 - 2 \sqrt{x + 5} = 3 x - 8 - x$

Subtract $6$ from both sides of the equation.

$- 2 \sqrt{x + 5} = 3 x - 8 - x - 6$

Subtract $x$ from $3 x$ .

$- 2 \sqrt{x + 5} = 2 x - 8 - 6$

Subtract $6$ from $- 8$ .

$- 2 \sqrt{x + 5} = 2 x - 14$

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

${(- 2 \sqrt{x + 5})}^{2} = {(2 x - 14)}^{2}$

Simplify each side of the equation.

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

${(- 2 {(x + 5)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

Simplify the left side.

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

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

${(- 2)}^{2} {({(x + 5)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

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

$4 {({(x + 5)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

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

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

$4 {(x + 5)}^{\frac{1}{2} \cdot 2} = {(2 x - 14)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x + 5)}^{\frac{1}{2} \cdot 2} = {(2 x - 14)}^{2}$

Rewrite the expression.

$4 {(x + 5)}^{1} = {(2 x - 14)}^{2}$

Simplify.

$4 (x + 5) = {(2 x - 14)}^{2}$

Apply the distributive property.

$4 x + 4 \cdot 5 = {(2 x - 14)}^{2}$

Multiply $4$ by $5$ .

$4 x + 20 = {(2 x - 14)}^{2}$

Simplify the right side.

Simplify ${(2 x - 14)}^{2}$ .

Rewrite ${(2 x - 14)}^{2}$ as $(2 x - 14) (2 x - 14)$ .

$4 x + 20 = (2 x - 14) (2 x - 14)$

Expand $(2 x - 14) (2 x - 14)$ using the FOIL Method.

Apply the distributive property.

$4 x + 20 = 2 x (2 x - 14) - 14 (2 x - 14)$

Apply the distributive property.

$4 x + 20 = 2 x (2 x) + 2 x \cdot - 14 - 14 (2 x - 14)$

Apply the distributive property.

$4 x + 20 = 2 x (2 x) + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 x + 20 = 2 \cdot 2 x \cdot x + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$4 x + 20 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Multiply $x$ by $x$ .

$4 x + 20 = 2 \cdot 2 x^{2} + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Multiply $2$ by $2$ .

$4 x + 20 = 4 x^{2} + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Multiply $- 14$ by $2$ .

$4 x + 20 = 4 x^{2} - 28 x - 14 (2 x) - 14 \cdot - 14$

Multiply $2$ by $- 14$ .

$4 x + 20 = 4 x^{2} - 28 x - 28 x - 14 \cdot - 14$

Multiply $- 14$ by $- 14$ .

$4 x + 20 = 4 x^{2} - 28 x - 28 x + 196$

Subtract $28 x$ from $- 28 x$ .

$4 x + 20 = 4 x^{2} - 56 x + 196$

Solve for $x$ .

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

$4 x^{2} - 56 x + 196 = 4 x + 20$

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

Subtract $4 x$ from both sides of the equation.

$4 x^{2} - 56 x + 196 - 4 x = 20$

Subtract $4 x$ from $- 56 x$ .

$4 x^{2} - 60 x + 196 = 20$

Subtract $20$ from both sides of the equation.

$4 x^{2} - 60 x + 196 - 20 = 0$

Subtract $20$ from $196$ .

$4 x^{2} - 60 x + 176 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 60 x + 176$ .

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 60 x + 176 = 0$

Factor $4$ out of $- 60 x$ .

$4 (x^{2}) + 4 (- 15 x) + 176 = 0$

Factor $4$ out of $176$ .

$4 x^{2} + 4 (- 15 x) + 4 \cdot 44 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 15 x)$ .

$4 (x^{2} - 15 x) + 4 \cdot 44 = 0$

Factor $4$ out of $4 (x^{2} - 15 x) + 4 \cdot 44$ .

$4 (x^{2} - 15 x + 44) = 0$

Factor.

Factor $x^{2} - 15 x + 44$ 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 $44$ and whose sum is $- 15$ .

$- 11, - 4$

Write the factored form using these integers.

$4 ((x - 11) (x - 4)) = 0$

Remove unnecessary parentheses.

$4 (x - 11) (x - 4) = 0$

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

$x - 11 = 0$

$x - 4 = 0$

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

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

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $4 (x - 11) (x - 4) = 0$ true.

$x = 11, 4$

Exclude the solutions that do not make $\sqrt{3 x - 8} + 1 = \sqrt{x + 5}$ true.

$x = 4$

Solve for x square root of 3x-8+1 = square root of x+5

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