Finite Math Examples

$\frac{3 \sqrt{x} + 2}{\sqrt{x} + 3}$

Step 1

Multiply $\frac{3 \sqrt{x} + 2}{\sqrt{x} + 3}$ by $\frac{\sqrt{x} - 3}{\sqrt{x} - 3}$ .

$\frac{3 \sqrt{x} + 2}{\sqrt{x} + 3} \cdot \frac{\sqrt{x} - 3}{\sqrt{x} - 3}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{3 \sqrt{x} + 2}{\sqrt{x} + 3}$ by $\frac{\sqrt{x} - 3}{\sqrt{x} - 3}$ .

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{(\sqrt{x} + 3) (\sqrt{x} - 3)}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{{\sqrt{x}}^{2} + \sqrt{x} \cdot - 3 + 3 \sqrt{x} - 9}$

Step 2.3

Simplify.

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Step 2.3.1

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

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{{(x^{\frac{1}{2}})}^{2} - 9}$

Step 2.3.2

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

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{x^{\frac{1}{2} \cdot 2} - 9}$

Step 2.3.3

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

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{x^{\frac{2}{2}} - 9}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{x^{\frac{2}{2}} - 9}$

Step 2.3.4.2

Rewrite the expression.

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{x^{1} - 9}$

Step 2.3.5

Simplify.

$\frac{(3 \sqrt{x} + 2) (\sqrt{x} - 3)}{x - 9}$

Step 3

Expand $(3 \sqrt{x} + 2) (\sqrt{x} - 3)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{3 \sqrt{x} (\sqrt{x} - 3) + 2 (\sqrt{x} - 3)}{x - 9}$

Step 3.2

Apply the distributive property.

$\frac{3 \sqrt{x} \sqrt{x} + 3 \sqrt{x} \cdot - 3 + 2 (\sqrt{x} - 3)}{x - 9}$

Step 3.3

Apply the distributive property.

$\frac{3 \sqrt{x} \sqrt{x} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $3 \sqrt{x} \sqrt{x}$ .

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Step 4.1.1.1

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

$\frac{3 ({\sqrt{x}}^{1} \sqrt{x}) + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.1.2

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

$\frac{3 ({\sqrt{x}}^{1} {\sqrt{x}}^{1}) + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.1.3

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

$\frac{3 {\sqrt{x}}^{1 + 1} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.1.4

Add $1$ and $1$ .

$\frac{3 {\sqrt{x}}^{2} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.2

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

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Step 4.1.2.1

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

$\frac{3 {(x^{\frac{1}{2}})}^{2} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.2.2

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

$\frac{3 x^{\frac{1}{2} \cdot 2} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.2.3

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

$\frac{3 x^{\frac{2}{2}} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

$\frac{3 x^{\frac{2}{2}} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.2.4.2

Rewrite the expression.

$\frac{3 x^{1} + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.2.5

Simplify.

$\frac{3 x + 3 \sqrt{x} \cdot - 3 + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.3

Multiply $- 3$ by $3$ .

$\frac{3 x - 9 \sqrt{x} + 2 \sqrt{x} + 2 \cdot - 3}{x - 9}$

Step 4.1.4

Multiply $2$ by $- 3$ .

$\frac{3 x - 9 \sqrt{x} + 2 \sqrt{x} - 6}{x - 9}$

Step 4.2

Add $- 9 \sqrt{x}$ and $2 \sqrt{x}$ .

$\frac{3 x - 7 \sqrt{x} - 6}{x - 9}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{3 x - 7 x^{\frac{1}{2}} - 6}{x - 9}$

Step 5.2

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

$\frac{3 {(x^{\frac{1}{2}})}^{2} - 7 x^{\frac{1}{2}} - 6}{x - 9}$

Step 5.3

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

$\frac{3 u^{2} - 7 u - 6}{x - 9}$

Step 5.4

Factor by grouping.

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Step 5.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 6 = - 18$ and whose sum is $b = - 7$ .

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Step 5.4.1.1

Factor $- 7$ out of $- 7 u$ .

$\frac{3 u^{2} - 7 (u) - 6}{x - 9}$

Step 5.4.1.2

Rewrite $- 7$ as $2$ plus $- 9$

$\frac{3 u^{2} + (2 - 9) u - 6}{x - 9}$

Step 5.4.1.3

Apply the distributive property.

$\frac{3 u^{2} + 2 u - 9 u - 6}{x - 9}$

Step 5.4.2

Factor out the greatest common factor from each group.

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Step 5.4.2.1

Group the first two terms and the last two terms.

$\frac{(3 u^{2} + 2 u) - 9 u - 6}{x - 9}$

Step 5.4.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{u (3 u + 2) - 3 (3 u + 2)}{x - 9}$

Step 5.4.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 2$ .

$\frac{(3 u + 2) (u - 3)}{x - 9}$

Step 5.5

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

$\frac{(3 x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 3)}{x - 9}$