Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Multiply $\frac{3}{2 x - 3}$ by $\frac{2 x - 3}{2 x - 3}$ .

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

Step 2.2

Raise $2 x - 3$ to the power of $1$ .

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

Step 2.3

Raise $2 x - 3$ to the power of $1$ .

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

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.5

Add $1$ and $1$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $2$ by $3$ .

$\frac{6 x + 3 \cdot - 3 - 1}{{(2 x - 3)}^{2}}$

Step 4.3

Multiply $3$ by $- 3$ .

$\frac{6 x - 9 - 1}{{(2 x - 3)}^{2}}$

Step 4.4

Subtract $1$ from $- 9$ .

$\frac{6 x - 10}{{(2 x - 3)}^{2}}$

Step 4.5

Factor $2$ out of $6 x - 10$ .

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Step 4.5.1

Factor $2$ out of $6 x$ .

$\frac{2 (3 x) - 10}{{(2 x - 3)}^{2}}$

Step 4.5.2

Factor $2$ out of $- 10$ .

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

Step 4.5.3

Factor $2$ out of $2 (3 x) + 2 (- 5)$ .

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