Algebra Examples

$\frac{\sqrt{x} + 7}{\sqrt{x} + 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1

Simplify $\frac{\sqrt{x} + 7}{\sqrt{x} + 1}$ .

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Step 1.1

Multiply $\frac{\sqrt{x} + 7}{\sqrt{x} + 1}$ by $\frac{\sqrt{x} - 1}{\sqrt{x} - 1}$ .

$\frac{\sqrt{x} + 7}{\sqrt{x} + 1} \cdot \frac{\sqrt{x} - 1}{\sqrt{x} - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.2

Combine fractions.

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Step 1.2.1

Multiply $\frac{\sqrt{x} + 7}{\sqrt{x} + 1}$ by $\frac{\sqrt{x} - 1}{\sqrt{x} - 1}$ .

$\frac{(\sqrt{x} + 7) (\sqrt{x} - 1)}{(\sqrt{x} + 1) (\sqrt{x} - 1)} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.2.2

Expand the denominator using the FOIL method.

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

Step 1.2.3

Simplify.

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.3.4

Cancel the common factor of $2$ .

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Step 1.2.3.4.1

Cancel the common factor.

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

Step 1.2.3.4.2

Rewrite the expression.

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

Step 1.2.3.5

Simplify.

$\frac{(\sqrt{x} + 7) (\sqrt{x} - 1)}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.3

Expand $(\sqrt{x} + 7) (\sqrt{x} - 1)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$\frac{\sqrt{x} (\sqrt{x} - 1) + 7 (\sqrt{x} - 1)}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.3.2

Apply the distributive property.

$\frac{\sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 1 + 7 (\sqrt{x} - 1)}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.3.3

Apply the distributive property.

$\frac{\sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

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

$\frac{{\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.1.2

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

$\frac{{\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.1.3

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

$\frac{{\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.1.4

Add $1$ and $1$ .

$\frac{{\sqrt{x}}^{2} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.2

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

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Step 1.4.1.2.1

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

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

Step 1.4.1.2.2

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

$\frac{x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.2.3

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

$\frac{x^{\frac{2}{2}} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.2.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.4.1

Cancel the common factor.

$\frac{x^{\frac{2}{2}} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.2.4.2

Rewrite the expression.

$\frac{x^{1} + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.2.5

Simplify.

$\frac{x + \sqrt{x} \cdot - 1 + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.3

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

$\frac{x - 1 \cdot \sqrt{x} + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.4

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

$\frac{x - \sqrt{x} + 7 \sqrt{x} + 7 \cdot - 1}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.1.5

Multiply $7$ by $- 1$ .

$\frac{x - \sqrt{x} + 7 \sqrt{x} - 7}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.4.2

Add $- \sqrt{x}$ and $7 \sqrt{x}$ .

$\frac{x + 6 \sqrt{x} - 7}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

$\frac{x + 6 x^{\frac{1}{2}} - 7}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.5.2

Rewrite $x$ as ${(x^{\frac{1}{2}})}^{2}$ .

$\frac{{(x^{\frac{1}{2}})}^{2} + 6 x^{\frac{1}{2}} - 7}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.5.3

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$\frac{u^{2} + 6 u - 7}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.5.4

Factor $u^{2} + 6 u - 7$ using the AC method.

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Step 1.5.4.1

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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 1.5.4.2

Write the factored form using these integers.

$\frac{(u - 1) (u + 7)}{x - 1} = \frac{\sqrt{x} + 1}{\sqrt{x} - 1}$

Step 1.5.5

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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

Step 2

Simplify $\frac{\sqrt{x} + 1}{\sqrt{x} - 1}$ .

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Step 2.1

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

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

Step 2.2

Combine fractions.

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Step 2.2.1

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

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

Step 2.2.2

Expand the denominator using the FOIL method.

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Cancel the common factor of $2$ .

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Step 2.2.3.4.1

Cancel the common factor.

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

Step 2.2.3.4.2

Rewrite the expression.

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

Step 2.2.3.5

Simplify.

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 4$

Step 4