Solve for x square root of x-7+ square root of x+8=5

$\sqrt{x - 7} + \sqrt{x + 8} = 5$

Step 1

Subtract $\sqrt{x + 8}$ from both sides of the equation.

$\sqrt{x - 7} = 5 - \sqrt{x + 8}$

Step 2

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

${\sqrt{x - 7}}^{2} = {(5 - \sqrt{x + 8})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 7}$ as ${(x - 7)}^{\frac{1}{2}}$ .

${({(x - 7)}^{\frac{1}{2}})}^{2} = {(5 - \sqrt{x + 8})}^{2}$

Simplify the left side.

Simplify ${({(x - 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 7)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 7)}^{\frac{1}{2} \cdot 2} = {(5 - \sqrt{x + 8})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 7)}^{\frac{1}{2} \cdot 2} = {(5 - \sqrt{x + 8})}^{2}$

Rewrite the expression.

${(x - 7)}^{1} = {(5 - \sqrt{x + 8})}^{2}$

Simplify.

$x - 7 = {(5 - \sqrt{x + 8})}^{2}$

Simplify the right side.

Simplify ${(5 - \sqrt{x + 8})}^{2}$ .

Rewrite ${(5 - \sqrt{x + 8})}^{2}$ as $(5 - \sqrt{x + 8}) (5 - \sqrt{x + 8})$ .

$x - 7 = (5 - \sqrt{x + 8}) (5 - \sqrt{x + 8})$

Expand $(5 - \sqrt{x + 8}) (5 - \sqrt{x + 8})$ using the FOIL Method.

Apply the distributive property.

$x - 7 = 5 (5 - \sqrt{x + 8}) - \sqrt{x + 8} (5 - \sqrt{x + 8})$

Apply the distributive property.

$x - 7 = 5 \cdot 5 + 5 (- \sqrt{x + 8}) - \sqrt{x + 8} (5 - \sqrt{x + 8})$

Apply the distributive property.

$x - 7 = 5 \cdot 5 + 5 (- \sqrt{x + 8}) - \sqrt{x + 8} \cdot 5 - \sqrt{x + 8} (- \sqrt{x + 8})$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$x - 7 = 25 + 5 (- \sqrt{x + 8}) - \sqrt{x + 8} \cdot 5 - \sqrt{x + 8} (- \sqrt{x + 8})$

Multiply $- 1$ by $5$ .

$x - 7 = 25 - 5 \sqrt{x + 8} - \sqrt{x + 8} \cdot 5 - \sqrt{x + 8} (- \sqrt{x + 8})$

Multiply $5$ by $- 1$ .

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} - \sqrt{x + 8} (- \sqrt{x + 8})$

Multiply $- \sqrt{x + 8} (- \sqrt{x + 8})$ .

Multiply $- 1$ by $- 1$ .

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + 1 \sqrt{x + 8} \sqrt{x + 8}$

Multiply $\sqrt{x + 8}$ by $1$ .

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + \sqrt{x + 8} \sqrt{x + 8}$

Raise $\sqrt{x + 8}$ to the power of $1$ .

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {\sqrt{x + 8}}^{1} \sqrt{x + 8}$

Raise $\sqrt{x + 8}$ to the power of $1$ .

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {\sqrt{x + 8}}^{1} {\sqrt{x + 8}}^{1}$

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

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {\sqrt{x + 8}}^{1 + 1}$

Add $1$ and $1$ .

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {\sqrt{x + 8}}^{2}$

Rewrite ${\sqrt{x + 8}}^{2}$ as $x + 8$ .

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

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {({(x + 8)}^{\frac{1}{2}})}^{2}$

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

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {(x + 8)}^{\frac{1}{2} \cdot 2}$

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

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {(x + 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {(x + 8)}^{\frac{2}{2}}$

Rewrite the expression.

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + {(x + 8)}^{1}$

Simplify.

$x - 7 = 25 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + x + 8$

Add $25$ and $8$ .

$x - 7 = 33 - 5 \sqrt{x + 8} - 5 \sqrt{x + 8} + x$

Subtract $5 \sqrt{x + 8}$ from $- 5 \sqrt{x + 8}$ .

$x - 7 = 33 - 10 \sqrt{x + 8} + x$

Step 4

Solve for $- 10 \sqrt{x + 8}$ .

Rewrite the equation as $33 - 10 \sqrt{x + 8} + x = x - 7$ .

$33 - 10 \sqrt{x + 8} + x = x - 7$

Move all terms not containing $- 10 \sqrt{x + 8}$ to the right side of the equation.

Subtract $33$ from both sides of the equation.

$- 10 \sqrt{x + 8} + x = x - 7 - 33$

Subtract $x$ from both sides of the equation.

$- 10 \sqrt{x + 8} = x - 7 - 33 - x$

Combine the opposite terms in $x - 7 - 33 - x$ .

Subtract $x$ from $x$ .

$- 10 \sqrt{x + 8} = 0 - 7 - 33$

Subtract $7$ from $0$ .

$- 10 \sqrt{x + 8} = - 7 - 33$

Subtract $33$ from $- 7$ .

$- 10 \sqrt{x + 8} = - 40$

Step 5

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

${(- 10 \sqrt{x + 8})}^{2} = {(- 40)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 10 {(x + 8)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

Simplify the left side.

Simplify ${(- 10 {(x + 8)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 10 {(x + 8)}^{\frac{1}{2}}$ .

${(- 10)}^{2} {({(x + 8)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

Raise $- 10$ to the power of $2$ .

$100 {({(x + 8)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

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

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

$100 {(x + 8)}^{\frac{1}{2} \cdot 2} = {(- 40)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 {(x + 8)}^{\frac{1}{2} \cdot 2} = {(- 40)}^{2}$

Rewrite the expression.

$100 {(x + 8)}^{1} = {(- 40)}^{2}$

Simplify.

$100 (x + 8) = {(- 40)}^{2}$

Apply the distributive property.

$100 x + 100 \cdot 8 = {(- 40)}^{2}$

Multiply $100$ by $8$ .

$100 x + 800 = {(- 40)}^{2}$

Simplify the right side.

Raise $- 40$ to the power of $2$ .

$100 x + 800 = 1600$

Step 7

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $800$ from both sides of the equation.

$100 x = 1600 - 800$

Subtract $800$ from $1600$ .

$100 x = 800$

Divide each term in $100 x = 800$ by $100$ and simplify.

Divide each term in $100 x = 800$ by $100$ .

$\frac{100 x}{100} = \frac{800}{100}$

Simplify the left side.

Cancel the common factor of $100$ .

Cancel the common factor.

$\frac{100 x}{100} = \frac{800}{100}$

Divide $x$ by $1$ .

$x = \frac{800}{100}$

Simplify the right side.

Divide $800$ by $100$ .

$x = 8$

Solve for x square root of x-7+ square root of x+8=5

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