Solve for x square root of x plus four plus square root of x minus one equals five

$\sqrt{x + 4} + \sqrt{x - 1} = 5$

Subtract $\sqrt{x - 1}$ from both sides of the equation.

$\sqrt{x + 4} = 5 - \sqrt{x - 1}$

To reduce the radical in the left part of the equation, square both sides of the equation.

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

Clarify each side of the equation.

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First, multiply the exponents in ${({(x + 4)}^{\frac{1}{2}})}^{2}$ .

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

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

The common factor should be canceled of $2$ .

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Cancel the common factor.

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

Reformulate the expression.

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

Simplify.

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

Calculate $x$ .

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

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Should be rewritten ${(5 - \sqrt{x - 1})}^{2}$ as $(5 - \sqrt{x - 1}) (5 - \sqrt{x - 1})$ .

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

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

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Use the distributive law.

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

Apply the distributive property.

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

Apply the distributive property.

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

Clarify and Add like terms.

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Clarify each term.

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Multiply $5$ by $5$ .

$x + 4 = 25 + 5 (- \sqrt{x - 1}) - \sqrt{x - 1} \cdot 5 - \sqrt{x - 1} (- \sqrt{x - 1})$

Multiply $- 1$ by $5$ .

$x + 4 = 25 - 5 \sqrt{x - 1} - \sqrt{x - 1} \cdot 5 - \sqrt{x - 1} (- \sqrt{x - 1})$

Multiply $5$ by $- 1$ .

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} - \sqrt{x - 1} (- \sqrt{x - 1})$

Multiply $- \sqrt{x - 1} (- \sqrt{x - 1})$ .

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Multiply $- 1$ by $- 1$ .

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + 1 \sqrt{x - 1} \sqrt{x - 1}$

Multiply $\sqrt{x - 1}$ by $1$ .

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + \sqrt{x - 1} \sqrt{x - 1}$

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

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {\sqrt{x - 1}}^{1} \sqrt{x - 1}$

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

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {\sqrt{x - 1}}^{1} {\sqrt{x - 1}}^{1}$

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

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {\sqrt{x - 1}}^{1 + 1}$

Add $1$ and $1$ .

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {\sqrt{x - 1}}^{2}$

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

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

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {({(x - 1)}^{\frac{1}{2}})}^{2}$

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

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {(x - 1)}^{\frac{1}{2} \cdot 2}$

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

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {(x - 1)}^{\frac{2}{2}}$

The common factor $2$ should be canceled.

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Cancel the common factor.

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {(x - 1)}^{\frac{2}{2}}$

First, divide $1$ by $1$ .

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + {(x - 1)}^{1}$

Simplify.

$x + 4 = 25 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + x - 1$

Subtract $1$ from $25$ .

$x + 4 = 24 - 5 \sqrt{x - 1} - 5 \sqrt{x - 1} + x$

Subtract $5 \sqrt{x - 1}$ from $- 5 \sqrt{x - 1}$ .

$x + 4 = 24 - 10 \sqrt{x - 1} + x$

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

$24 - 10 \sqrt{x - 1} + x = x + 4$

Move all terms not containing $\sqrt{x - 1}$ to the right side of the equation.

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Subtract $24$ from both sides of the equation.

$- 10 \sqrt{x - 1} + x = x + 4 - 24$

Subtract $x$ from both sides of the equation.

$- 10 \sqrt{x - 1} = x + 4 - 24 - x$

Combine the opposite terms in $x + 4 - 24 - x$ .

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Subtract $x$ from $x$ .

$- 10 \sqrt{x - 1} = 0 + 4 - 24$

Sum $0$ and $4$ .

$- 10 \sqrt{x - 1} = 4 - 24$

Subtract $24$ from $4$ .

$- 10 \sqrt{x - 1} = - 20$

Divide each term by $- 10$ and simplify.

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Divide each term in $- 10 \sqrt{x - 1} = - 20$ by $- 10$ .

$\frac{- 10 \sqrt{x - 1}}{- 10} = \frac{- 20}{- 10}$

Cancel the common factor of $- 10$ .

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Cancel the common factor.

$\frac{- 10 \sqrt{x - 1}}{- 10} = \frac{- 20}{- 10}$

Divide $\sqrt{x - 1}$ by $1$ .

$\sqrt{x - 1} = \frac{- 20}{- 10}$

Divide $- 20$ by $- 10$ .

$\sqrt{x - 1} = 2$

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

${\sqrt{x - 1}}^{2} = 2^{2}$

Simplify each side of the equation.

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

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

${(x - 1)}^{\frac{1}{2} \cdot 2} = 2^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

${(x - 1)}^{\frac{1}{2} \cdot 2} = 2^{2}$

Rewrite the expression.

${(x - 1)}^{1} = 2^{2}$

Simplify.

$x - 1 = 2^{2}$

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

$x - 1 = 4$

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

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Add $1$ to both sides of the equation.

$x = 4 + 1$

Add $4$ and $1$ .

$x = 5$

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