Basic Math Examples

$\frac{4 d^{2} - 1}{d^{2} - 9} \div \frac{10 d + 5}{d + 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 d^{2} - 1}{d^{2} - 9} \cdot \frac{d + 3}{10 d + 5}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(2 d)}^{2} - 1}{d^{2} - 9} \cdot \frac{d + 3}{10 d + 5}$

Step 2.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(2 d)}^{2} - 1^{2}}{d^{2} - 9} \cdot \frac{d + 3}{10 d + 5}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 d$ and $b = 1$ .

$\frac{(2 d + 1) (2 d - 1)}{d^{2} - 9} \cdot \frac{d + 3}{10 d + 5}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{(2 d + 1) (2 d - 1)}{d^{2} - 3^{2}} \cdot \frac{d + 3}{10 d + 5}$

Step 3.2

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

$\frac{(2 d + 1) (2 d - 1)}{(d + 3) (d - 3)} \cdot \frac{d + 3}{10 d + 5}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $d + 3$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(2 d + 1) (2 d - 1)}{(d + 3) (d - 3)} \cdot \frac{d + 3}{10 d + 5}$

Step 4.1.2

Rewrite the expression.

$\frac{(2 d + 1) (2 d - 1)}{d - 3} \cdot \frac{1}{10 d + 5}$

Step 4.2

Multiply $\frac{(2 d + 1) (2 d - 1)}{d - 3}$ by $\frac{1}{10 d + 5}$ .

$\frac{(2 d + 1) (2 d - 1)}{(d - 3) (10 d + 5)}$

Step 5

Factor $5$ out of $10 d + 5$ .

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Step 5.1

Factor $5$ out of $10 d$ .

$\frac{(2 d + 1) (2 d - 1)}{(d - 3) (5 (2 d) + 5)}$

Step 5.2

Factor $5$ out of $5$ .

$\frac{(2 d + 1) (2 d - 1)}{(d - 3) (5 (2 d) + 5 (1))}$

Step 5.3

Factor $5$ out of $5 (2 d) + 5 (1)$ .

$\frac{(2 d + 1) (2 d - 1)}{(d - 3) (5 (2 d + 1))}$

$\frac{(2 d + 1) (2 d - 1)}{(d - 3) \cdot 5 (2 d + 1)}$

Step 6

Cancel the common factor of $2 d + 1$ .

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Step 6.1

Cancel the common factor.

$\frac{(2 d + 1) (2 d - 1)}{(d - 3) \cdot 5 (2 d + 1)}$

Step 6.2

Rewrite the expression.

$\frac{2 d - 1}{(d - 3) \cdot 5}$

Step 7

Move $5$ to the left of $d - 3$ .

$\frac{2 d - 1}{5 (d - 3)}$