Algebra Examples

$y = \frac{3 x^{2} - 12}{8 x^{2} - 18 x + 4}$

Step 1

Factor $3$ out of $3 x^{2} - 12$ .

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Step 1.1

Factor $3$ out of $3 x^{2}$ .

$y = \frac{3 (x^{2}) - 12}{8 x^{2} - 18 x + 4}$

Step 1.2

Factor $3$ out of $- 12$ .

$y = \frac{3 x^{2} + 3 \cdot - 4}{8 x^{2} - 18 x + 4}$

Step 1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 4$ .

$y = \frac{3 (x^{2} - 4)}{8 x^{2} - 18 x + 4}$

Step 2

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

$y = \frac{3 (x^{2} - 2^{2})}{8 x^{2} - 18 x + 4}$

Step 3

Factor.

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Step 3.1

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

$y = \frac{3 ((x + 2) (x - 2))}{8 x^{2} - 18 x + 4}$

Step 3.2

Remove unnecessary parentheses.

$y = \frac{3 (x + 2) (x - 2)}{8 x^{2} - 18 x + 4}$

Step 4

Factor $2$ out of $8 x^{2} - 18 x + 4$ .

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Step 4.1

Factor $2$ out of $8 x^{2}$ .

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2}) - 18 x + 4}$

Step 4.2

Factor $2$ out of $- 18 x$ .

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2}) + 2 (- 9 x) + 4}$

Step 4.3

Factor $2$ out of $4$ .

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2}) + 2 (- 9 x) + 2 (2)}$

Step 4.4

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

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2} - 9 x) + 2 (2)}$

Step 4.5

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

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2} - 9 x + 2)}$

Step 5

Factor.

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Step 5.1

Factor by grouping.

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Step 5.1.1

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 $a \cdot c = 4 \cdot 2 = 8$ and whose sum is $b = - 9$ .

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Step 5.1.1.1

Factor $- 9$ out of $- 9 x$ .

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2} - 9 (x) + 2)}$

Step 5.1.1.2

Rewrite $- 9$ as $- 1$ plus $- 8$

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2} + (- 1 - 8) x + 2)}$

Step 5.1.1.3

Apply the distributive property.

$y = \frac{3 (x + 2) (x - 2)}{2 (4 x^{2} - 1 x - 8 x + 2)}$

Step 5.1.2

Factor out the greatest common factor from each group.

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Step 5.1.2.1

Group the first two terms and the last two terms.

$y = \frac{3 (x + 2) (x - 2)}{2 ((4 x^{2} - 1 x) - 8 x + 2)}$

Step 5.1.2.2

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

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

Step 5.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

Reduce the expression $\frac{3 (x + 2) (x - 2)}{2 (4 x - 1) (x - 2)}$ by cancelling the common factors.

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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