Precalculus Examples

$\frac{5 x^{2} + 9 x - 18}{2 x^{2} - 4 x - 30}$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 5 \cdot - 18 = - 90$ and whose sum is $b = 9$ .

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Step 1.1.1

Factor $9$ out of $9 x$ .

$\frac{5 x^{2} + 9 (x) - 18}{2 x^{2} - 4 x - 30}$

Step 1.1.2

Rewrite $9$ as $- 6$ plus $15$

$\frac{5 x^{2} + (- 6 + 15) x - 18}{2 x^{2} - 4 x - 30}$

Step 1.1.3

Apply the distributive property.

$\frac{5 x^{2} - 6 x + 15 x - 18}{2 x^{2} - 4 x - 30}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(5 x^{2} - 6 x) + 15 x - 18}{2 x^{2} - 4 x - 30}$

Step 1.2.2

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

$\frac{x (5 x - 6) + 3 (5 x - 6)}{2 x^{2} - 4 x - 30}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 6$ .

$\frac{(5 x - 6) (x + 3)}{2 x^{2} - 4 x - 30}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2$ out of $2 x^{2} - 4 x - 30$ .

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Step 2.1.1

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

$\frac{(5 x - 6) (x + 3)}{2 (x^{2}) - 4 x - 30}$

Step 2.1.2

Factor $2$ out of $- 4 x$ .

$\frac{(5 x - 6) (x + 3)}{2 (x^{2}) + 2 (- 2 x) - 30}$

Step 2.1.3

Factor $2$ out of $- 30$ .

$\frac{(5 x - 6) (x + 3)}{2 x^{2} + 2 (- 2 x) + 2 \cdot - 15}$

Step 2.1.4

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

$\frac{(5 x - 6) (x + 3)}{2 (x^{2} - 2 x) + 2 \cdot - 15}$

Step 2.1.5

Factor $2$ out of $2 (x^{2} - 2 x) + 2 \cdot - 15$ .

$\frac{(5 x - 6) (x + 3)}{2 (x^{2} - 2 x - 15)}$

Step 2.2

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 2.2.2

Write the factored form using these integers.

$\frac{(5 x - 6) (x + 3)}{2 ((x - 5) (x + 3))}$

$\frac{(5 x - 6) (x + 3)}{2 (x - 5) (x + 3)}$

Step 3

Cancel the common factor of $x + 3$ .

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Step 3.1

Cancel the common factor.

$\frac{(5 x - 6) (x + 3)}{2 (x - 5) (x + 3)}$

Step 3.2

Rewrite the expression.

$\frac{5 x - 6}{2 (x - 5)}$