Algebra Examples

$\frac{16 - 9 a^{2}}{9 a^{2} - 24 a + 16}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $16$ as $4^{2}$ .

$\frac{4^{2} - 9 a^{2}}{9 a^{2} - 24 a + 16}$

Step 1.2

Rewrite $9 a^{2}$ as ${(3 a)}^{2}$ .

$\frac{4^{2} - {(3 a)}^{2}}{9 a^{2} - 24 a + 16}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 3 a$ .

$\frac{(4 + 3 a) (4 - (3 a))}{9 a^{2} - 24 a + 16}$

Step 1.4

Multiply $3$ by $- 1$ .

$\frac{(4 + 3 a) (4 - 3 a)}{9 a^{2} - 24 a + 16}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9 a^{2}$ as ${(3 a)}^{2}$ .

$\frac{(4 + 3 a) (4 - 3 a)}{{(3 a)}^{2} - 24 a + 16}$

Step 2.2

Rewrite $16$ as $4^{2}$ .

$\frac{(4 + 3 a) (4 - 3 a)}{{(3 a)}^{2} - 24 a + 4^{2}}$

Step 2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$24 a = 2 \cdot (3 a) \cdot 4$

Step 2.4

Rewrite the polynomial.

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

Step 2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 a$ and $b = 4$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $4 - 3 a$ and ${(3 a - 4)}^{2}$ .

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Step 3.1.1

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 3.1.2

Factor $- 1$ out of $- 3 a$ .

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

Step 3.1.3

Factor $- 1$ out of $- 1 (- 4) - (3 a)$ .

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

Step 3.1.4

Reorder terms.

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

Step 3.1.5

Factor $3 a - 4$ out of $(4 + 3 a) (- 1 (3 a - 4))$ .

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

Step 3.1.6

Cancel the common factors.

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Step 3.1.6.1

Factor $3 a - 4$ out of ${(3 a - 4)}^{2}$ .

$\frac{(3 a - 4) ((4 + 3 a) \cdot (- 1))}{(3 a - 4) (3 a - 4)}$

Step 3.1.6.2

Cancel the common factor.

$\frac{(3 a - 4) ((4 + 3 a) \cdot (- 1))}{(3 a - 4) (3 a - 4)}$

Step 3.1.6.3

Rewrite the expression.

$\frac{(4 + 3 a) \cdot (- 1)}{3 a - 4}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $- 1$ to the left of $4 + 3 a$ .

$\frac{- 1 \cdot (4 + 3 a)}{3 a - 4}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{4 + 3 a}{3 a - 4}$