Basic Math Examples

$\frac{4 a^{2} c^{2} - 9 a^{2}}{8 a^{2} c^{2} - 18 a^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $a^{2}$ out of $4 a^{2} c^{2} - 9 a^{2}$ .

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Step 1.1.1

Factor $a^{2}$ out of $4 a^{2} c^{2}$ .

$\frac{a^{2} (4 c^{2}) - 9 a^{2}}{8 a^{2} c^{2} - 18 a^{2}}$

Step 1.1.2

Factor $a^{2}$ out of $- 9 a^{2}$ .

$\frac{a^{2} (4 c^{2}) + a^{2} \cdot - 9}{8 a^{2} c^{2} - 18 a^{2}}$

Step 1.1.3

Factor $a^{2}$ out of $a^{2} (4 c^{2}) + a^{2} \cdot - 9$ .

$\frac{a^{2} (4 c^{2} - 9)}{8 a^{2} c^{2} - 18 a^{2}}$

Step 1.2

Rewrite $4 c^{2}$ as ${(2 c)}^{2}$ .

$\frac{a^{2} ({(2 c)}^{2} - 9)}{8 a^{2} c^{2} - 18 a^{2}}$

Step 1.3

Rewrite $9$ as $3^{2}$ .

$\frac{a^{2} ({(2 c)}^{2} - 3^{2})}{8 a^{2} c^{2} - 18 a^{2}}$

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 c$ and $b = 3$ .

$\frac{a^{2} (2 c + 3) (2 c - 3)}{8 a^{2} c^{2} - 18 a^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2 a^{2}$ out of $8 a^{2} c^{2} - 18 a^{2}$ .

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Step 2.1.1

Factor $2 a^{2}$ out of $8 a^{2} c^{2}$ .

$\frac{a^{2} (2 c + 3) (2 c - 3)}{2 a^{2} (4 c^{2}) - 18 a^{2}}$

Step 2.1.2

Factor $2 a^{2}$ out of $- 18 a^{2}$ .

$\frac{a^{2} (2 c + 3) (2 c - 3)}{2 a^{2} (4 c^{2}) + 2 a^{2} (- 9)}$

Step 2.1.3

Factor $2 a^{2}$ out of $2 a^{2} (4 c^{2}) + 2 a^{2} (- 9)$ .

$\frac{a^{2} (2 c + 3) (2 c - 3)}{2 a^{2} (4 c^{2} - 9)}$

Step 2.2

Rewrite $4 c^{2}$ as ${(2 c)}^{2}$ .

$\frac{a^{2} (2 c + 3) (2 c - 3)}{2 a^{2} ({(2 c)}^{2} - 9)}$

Step 2.3

Rewrite $9$ as $3^{2}$ .

$\frac{a^{2} (2 c + 3) (2 c - 3)}{2 a^{2} ({(2 c)}^{2} - 3^{2})}$

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 c$ and $b = 3$ .

$\frac{a^{2} (2 c + 3) (2 c - 3)}{2 a^{2} (2 c + 3) (2 c - 3)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $a^{2}$ .

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Step 3.1.1

Cancel the common factor.

$\frac{a^{2} (2 c + 3) (2 c - 3)}{2 a^{2} (2 c + 3) (2 c - 3)}$

Step 3.1.2

Rewrite the expression.

$\frac{(2 c + 3) (2 c - 3)}{(2 (2 c + 3)) (2 c - 3)}$

Step 3.2

Cancel the common factor of $2 c + 3$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(2 c + 3) (2 c - 3)}{2 (2 c + 3) (2 c - 3)}$

Step 3.2.2

Rewrite the expression.

$\frac{2 c - 3}{(2) (2 c - 3)}$

Step 3.3

Cancel the common factor of $2 c - 3$ .

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Step 3.3.1

Cancel the common factor.

$\frac{2 c - 3}{2 (2 c - 3)}$

Step 3.3.2

Rewrite the expression.

$\frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$