Algebra Examples

$\sqrt{x + 15} = 5 + \sqrt{x}$

Step 1

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

${\sqrt{x + 15}}^{2} = {(5 + \sqrt{x})}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 15}$ as ${(x + 15)}^{\frac{1}{2}}$ .

${({(x + 15)}^{\frac{1}{2}})}^{2} = {(5 + \sqrt{x})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x + 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x + 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x + 15)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{x})}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x + 15)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{x})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 15)}^{1} = {(5 + \sqrt{x})}^{2}$

Step 2.2.1.2

Simplify.

$x + 15 = {(5 + \sqrt{x})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$x + 15 = (5 + \sqrt{x}) (5 + \sqrt{x})$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$x + 15 = 5 (5 + \sqrt{x}) + \sqrt{x} (5 + \sqrt{x})$

Step 2.3.1.2.2

Apply the distributive property.

$x + 15 = 5 \cdot 5 + 5 \sqrt{x} + \sqrt{x} (5 + \sqrt{x})$

Step 2.3.1.2.3

Apply the distributive property.

$x + 15 = 5 \cdot 5 + 5 \sqrt{x} + \sqrt{x} \cdot 5 + \sqrt{x} \sqrt{x}$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $5$ by $5$ .

$x + 15 = 25 + 5 \sqrt{x} + \sqrt{x} \cdot 5 + \sqrt{x} \sqrt{x}$

Step 2.3.1.3.1.2

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

$x + 15 = 25 + 5 \sqrt{x} + 5 \cdot \sqrt{x} + \sqrt{x} \sqrt{x}$

Step 2.3.1.3.1.3

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

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Step 2.3.1.3.1.3.1

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

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

Step 2.3.1.3.1.3.2

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

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

Step 2.3.1.3.1.3.3

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

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Step 2.3.1.3.1.3.4

Add $1$ and $1$ .

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{2}$

Step 2.3.1.3.1.4

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

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Step 2.3.1.3.1.4.1

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

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.1.4.2

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

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.1.4.3

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

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{\frac{2}{2}}$

Step 2.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.4.4.1

Cancel the common factor.

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{\frac{2}{2}}$

Step 2.3.1.3.1.4.4.2

Rewrite the expression.

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{1}$

Step 2.3.1.3.1.4.5

Simplify.

$x + 15 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x$

Step 2.3.1.3.2

Add $5 \sqrt{x}$ and $5 \sqrt{x}$ .

$x + 15 = 25 + 10 \sqrt{x} + x$

Step 3

Solve for $10 \sqrt{x}$ .

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Step 3.1

Rewrite the equation as $25 + 10 \sqrt{x} + x = x + 15$ .

$25 + 10 \sqrt{x} + x = x + 15$

Step 3.2

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

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Step 3.2.1

Subtract $25$ from both sides of the equation.

$10 \sqrt{x} + x = x + 15 - 25$

Step 3.2.2

Subtract $x$ from both sides of the equation.

$10 \sqrt{x} = x + 15 - 25 - x$

Step 3.2.3

Combine the opposite terms in $x + 15 - 25 - x$ .

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Step 3.2.3.1

Subtract $x$ from $x$ .

$10 \sqrt{x} = 0 + 15 - 25$

Step 3.2.3.2

Add $0$ and $15$ .

$10 \sqrt{x} = 15 - 25$

Step 3.2.4

Subtract $25$ from $15$ .

$10 \sqrt{x} = - 10$

Step 4

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

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

Step 5

Simplify each side of the equation.

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

$100 {(x^{\frac{1}{2}})}^{2} = {(- 10)}^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$100 x^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$100 x^{\frac{1}{2} \cdot 2} = {(- 10)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$100 x^{1} = {(- 10)}^{2}$

Step 5.2.1.4

Simplify.

$100 x = {(- 10)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$100 x = 100$

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $100$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $100$ by $100$ .

$x = 1$

Step 7

Exclude the solutions that do not make $\sqrt{x + 15} = 5 + \sqrt{x}$ true.

$No solution$