Algebra Examples

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

Step 1

Multiply $\frac{\sqrt{x} + 3}{2 \sqrt{x} - 1}$ by $\frac{2 \sqrt{x} + 1}{2 \sqrt{x} + 1}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{x} + 3}{2 \sqrt{x} - 1}$ by $\frac{2 \sqrt{x} + 1}{2 \sqrt{x} + 1}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{x} + 3) (2 \sqrt{x} + 1)}{4 {\sqrt{x}}^{2} + 2 \sqrt{x} - 2 \sqrt{x} - 1}$

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 $x^{\frac{1}{2}}$ .

$\frac{(\sqrt{x} + 3) (2 \sqrt{x} + 1)}{4 {(x^{\frac{1}{2}})}^{2} - 1}$

Step 2.3.2

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

$\frac{(\sqrt{x} + 3) (2 \sqrt{x} + 1)}{4 x^{\frac{1}{2} \cdot 2} - 1}$

Step 2.3.3

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

$\frac{(\sqrt{x} + 3) (2 \sqrt{x} + 1)}{4 x^{\frac{2}{2}} - 1}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\frac{(\sqrt{x} + 3) (2 \sqrt{x} + 1)}{4 x^{\frac{2}{2}} - 1}$

Step 2.3.4.2

Rewrite the expression.

$\frac{(\sqrt{x} + 3) (2 \sqrt{x} + 1)}{4 x^{1} - 1}$

Step 2.3.5

Simplify.

$\frac{(\sqrt{x} + 3) (2 \sqrt{x} + 1)}{4 x - 1}$

Step 3

Expand $(\sqrt{x} + 3) (2 \sqrt{x} + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{x} (2 \sqrt{x} + 1) + 3 (2 \sqrt{x} + 1)}{4 x - 1}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{x} (2 \sqrt{x}) + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x} + 1)}{4 x - 1}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{x} (2 \sqrt{x}) + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

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

Rewrite using the commutative property of multiplication.

$\frac{2 \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.2

Multiply $2 \sqrt{x} \sqrt{x}$ .

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Step 4.1.2.1

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

$\frac{2 ({\sqrt{x}}^{1} \sqrt{x}) + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.2.2

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

$\frac{2 ({\sqrt{x}}^{1} {\sqrt{x}}^{1}) + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.2.3

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

$\frac{2 {\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.2.4

Add $1$ and $1$ .

$\frac{2 {\sqrt{x}}^{2} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{2 {(x^{\frac{1}{2}})}^{2} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.3.2

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

$\frac{2 x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.3.3

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

$\frac{2 x^{\frac{2}{2}} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{2 x^{\frac{2}{2}} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{2 x^{1} + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.3.5

Simplify.

$\frac{2 x + \sqrt{x} \cdot 1 + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.4

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

$\frac{2 x + \sqrt{x} + 3 (2 \sqrt{x}) + 3 \cdot 1}{4 x - 1}$

Step 4.1.5

Multiply $2$ by $3$ .

$\frac{2 x + \sqrt{x} + 6 \sqrt{x} + 3 \cdot 1}{4 x - 1}$

Step 4.1.6

Multiply $3$ by $1$ .

$\frac{2 x + \sqrt{x} + 6 \sqrt{x} + 3}{4 x - 1}$

Step 4.2

Add $\sqrt{x}$ and $6 \sqrt{x}$ .

$\frac{2 x + 7 \sqrt{x} + 3}{4 x - 1}$

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 $x^{\frac{1}{2}}$ .

$\frac{2 x + 7 x^{\frac{1}{2}} + 3}{4 x - 1}$

Step 5.2

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

$\frac{2 {(x^{\frac{1}{2}})}^{2} + 7 x^{\frac{1}{2}} + 3}{4 x - 1}$

Step 5.3

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

$\frac{2 u^{2} + 7 u + 3}{4 x - 1}$

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = 7$ .

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Step 5.4.1.1

Factor $7$ out of $7 u$ .

$\frac{2 u^{2} + 7 (u) + 3}{4 x - 1}$

Step 5.4.1.2

Rewrite $7$ as $1$ plus $6$

$\frac{2 u^{2} + (1 + 6) u + 3}{4 x - 1}$

Step 5.4.1.3

Apply the distributive property.

$\frac{2 u^{2} + 1 u + 6 u + 3}{4 x - 1}$

Step 5.4.1.4

Multiply $u$ by $1$ .

$\frac{2 u^{2} + u + 6 u + 3}{4 x - 1}$

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.

$\frac{(2 u^{2} + u) + 6 u + 3}{4 x - 1}$

Step 5.4.2.2

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

$\frac{u (2 u + 1) + 3 (2 u + 1)}{4 x - 1}$

Step 5.4.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$\frac{(2 u + 1) (u + 3)}{4 x - 1}$

Step 5.5

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

$\frac{(2 x^{\frac{1}{2}} + 1) (x^{\frac{1}{2}} + 3)}{4 x - 1}$