Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $5$ by $5$ .

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

Step 3.3.1.3.1.2

Multiply $- 1$ by $5$ .

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

Step 3.3.1.3.1.3

Multiply $5$ by $- 1$ .

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

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.3.1.4.2

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

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

Step 3.3.1.3.1.4.3

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

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

Step 3.3.1.3.1.4.4

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

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

Step 3.3.1.3.1.4.5

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

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

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

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

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

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

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

Step 3.3.1.3.1.5.2

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

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

Step 3.3.1.3.1.5.3

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

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

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.5.5

Simplify.

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

Step 3.3.1.3.2

Subtract $1$ from $25$ .

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

Step 3.3.1.3.3

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

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

Step 4

Solve for $- 10 \sqrt{x - 1}$ .

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Step 4.1

Rewrite the equation as $24 - 10 \sqrt{x - 1} + x = x + 4$ .

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

Step 4.2

Move all terms not containing $- 10 \sqrt{x - 1}$ to the right side of the equation.

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Step 4.2.1

Subtract $24$ from both sides of the equation.

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

Step 4.2.2

Subtract $x$ from both sides of the equation.

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

Step 4.2.3

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

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Step 4.2.3.1

Subtract $x$ from $x$ .

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

Step 4.2.3.2

Add $0$ and $4$ .

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

Step 4.2.4

Subtract $24$ from $4$ .

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

Step 5

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

${(- 10 \sqrt{x - 1})}^{2} = {(- 20)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

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

${(- 10 {(x - 1)}^{\frac{1}{2}})}^{2} = {(- 20)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

${(- 10)}^{2} {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- 20)}^{2}$

Step 6.2.1.2

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

$100 {({(x - 1)}^{\frac{1}{2}})}^{2} = {(- 20)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$100 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(- 20)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$100 {(x - 1)}^{\frac{1}{2} \cdot 2} = {(- 20)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$100 {(x - 1)}^{1} = {(- 20)}^{2}$

Step 6.2.1.4

Simplify.

$100 (x - 1) = {(- 20)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$100 x + 100 \cdot - 1 = {(- 20)}^{2}$

Step 6.2.1.6

Multiply $100$ by $- 1$ .

$100 x - 100 = {(- 20)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $- 20$ to the power of $2$ .

$100 x - 100 = 400$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Add $100$ to both sides of the equation.

$100 x = 400 + 100$

Step 7.1.2

Add $400$ and $100$ .

$100 x = 500$

Step 7.2

Divide each term in $100 x = 500$ by $100$ and simplify.

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Step 7.2.1

Divide each term in $100 x = 500$ by $100$ .

$\frac{100 x}{100} = \frac{500}{100}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $100$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{100 x}{100} = \frac{500}{100}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{500}{100}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $500$ by $100$ .

$x = 5$