Pre-Algebra Examples

$\frac{3}{4 x^{2} - 21 x + 27} + \frac{1}{4 x^{2} - 17 x + 18}$

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $a \cdot c = 4 \cdot 27 = 108$ and whose sum is $b = - 21$ .

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Step 1.1.1.1

Factor $- 21$ out of $- 21 x$ .

$\frac{3}{4 x^{2} - 21 (x) + 27} + \frac{1}{4 x^{2} - 17 x + 18}$

Step 1.1.1.2

Rewrite $- 21$ as $- 9$ plus $- 12$

$\frac{3}{4 x^{2} + (- 9 - 12) x + 27} + \frac{1}{4 x^{2} - 17 x + 18}$

Step 1.1.1.3

Apply the distributive property.

$\frac{3}{4 x^{2} - 9 x - 12 x + 27} + \frac{1}{4 x^{2} - 17 x + 18}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{3}{(4 x^{2} - 9 x) - 12 x + 27} + \frac{1}{4 x^{2} - 17 x + 18}$

Step 1.1.2.2

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

$\frac{3}{x (4 x - 9) - 3 (4 x - 9)} + \frac{1}{4 x^{2} - 17 x + 18}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 9$ .

$\frac{3}{(4 x - 9) (x - 3)} + \frac{1}{4 x^{2} - 17 x + 18}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 4 \cdot 18 = 72$ and whose sum is $b = - 17$ .

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Step 1.2.1.1

Factor $- 17$ out of $- 17 x$ .

$\frac{3}{(4 x - 9) (x - 3)} + \frac{1}{4 x^{2} - 17 (x) + 18}$

Step 1.2.1.2

Rewrite $- 17$ as $- 8$ plus $- 9$

$\frac{3}{(4 x - 9) (x - 3)} + \frac{1}{4 x^{2} + (- 8 - 9) x + 18}$

Step 1.2.1.3

Apply the distributive property.

$\frac{3}{(4 x - 9) (x - 3)} + \frac{1}{4 x^{2} - 8 x - 9 x + 18}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{3}{(4 x - 9) (x - 3)} + \frac{1}{(4 x^{2} - 8 x) - 9 x + 18}$

Step 1.2.2.2

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

$\frac{3}{(4 x - 9) (x - 3)} + \frac{1}{4 x (x - 2) - 9 (x - 2)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$\frac{3}{(4 x - 9) (x - 3)} + \frac{1}{(x - 2) (4 x - 9)}$

Step 2

To write $\frac{3}{(4 x - 9) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{3}{(4 x - 9) (x - 3)} \cdot \frac{x - 2}{x - 2} + \frac{1}{(x - 2) (4 x - 9)}$

Step 3

To write $\frac{1}{(x - 2) (4 x - 9)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{3}{(4 x - 9) (x - 3)} \cdot \frac{x - 2}{x - 2} + \frac{1}{(x - 2) (4 x - 9)} \cdot \frac{x - 3}{x - 3}$

Step 4

Write each expression with a common denominator of $(4 x - 9) (x - 3) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3}{(4 x - 9) (x - 3)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{3 (x - 2)}{(4 x - 9) (x - 3) (x - 2)} + \frac{1}{(x - 2) (4 x - 9)} \cdot \frac{x - 3}{x - 3}$

Step 4.2

Multiply $\frac{1}{(x - 2) (4 x - 9)}$ by $\frac{x - 3}{x - 3}$ .

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

Step 4.3

Reorder the factors of $(4 x - 9) (x - 3) (x - 2)$ .

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

Step 4.4

Reorder the factors of $(x - 2) (4 x - 9) (x - 3)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{3 x + 3 \cdot - 2 + x - 3}{(4 x - 9) (x - 2) (x - 3)}$

Step 6.2

Multiply $3$ by $- 2$ .

$\frac{3 x - 6 + x - 3}{(4 x - 9) (x - 2) (x - 3)}$

Step 6.3

Add $3 x$ and $x$ .

$\frac{4 x - 6 - 3}{(4 x - 9) (x - 2) (x - 3)}$

Step 6.4

Subtract $3$ from $- 6$ .

$\frac{4 x - 9}{(4 x - 9) (x - 2) (x - 3)}$

Step 7

Cancel the common factor of $4 x - 9$ .

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Step 7.1

Cancel the common factor.

$\frac{4 x - 9}{(4 x - 9) (x - 2) (x - 3)}$

Step 7.2

Rewrite the expression.

$\frac{1}{(x - 2) (x - 3)}$