Precalculus Examples

$\frac{5 t - 1}{4 t} \cdot \frac{4 t^{2} + 3 t}{16 t^{2} - 9}$

Step 1

Factor $t$ out of $4 t^{2} + 3 t$ .

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Step 1.1

Factor $t$ out of $4 t^{2}$ .

$\frac{5 t - 1}{4 t} \cdot \frac{t (4 t) + 3 t}{16 t^{2} - 9}$

Step 1.2

Factor $t$ out of $3 t$ .

$\frac{5 t - 1}{4 t} \cdot \frac{t (4 t) + t \cdot 3}{16 t^{2} - 9}$

Step 1.3

Factor $t$ out of $t (4 t) + t \cdot 3$ .

$\frac{5 t - 1}{4 t} \cdot \frac{t (4 t + 3)}{16 t^{2} - 9}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16 t^{2}$ as ${(4 t)}^{2}$ .

$\frac{5 t - 1}{4 t} \cdot \frac{t (4 t + 3)}{{(4 t)}^{2} - 9}$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$\frac{5 t - 1}{4 t} \cdot \frac{t (4 t + 3)}{{(4 t)}^{2} - 3^{2}}$

Step 2.3

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

$\frac{5 t - 1}{4 t} \cdot \frac{t (4 t + 3)}{(4 t + 3) (4 t - 3)}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $t$ .

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Step 3.1.1

Factor $t$ out of $4 t$ .

$\frac{5 t - 1}{t \cdot 4} \cdot \frac{t (4 t + 3)}{(4 t + 3) (4 t - 3)}$

Step 3.1.2

Cancel the common factor.

$\frac{5 t - 1}{t \cdot 4} \cdot \frac{t (4 t + 3)}{(4 t + 3) (4 t - 3)}$

Step 3.1.3

Rewrite the expression.

$\frac{5 t - 1}{4} \cdot \frac{4 t + 3}{(4 t + 3) (4 t - 3)}$

Step 3.2

Multiply $\frac{5 t - 1}{4}$ by $\frac{4 t + 3}{(4 t + 3) (4 t - 3)}$ .

$\frac{(5 t - 1) (4 t + 3)}{4 ((4 t + 3) (4 t - 3))}$

Step 3.3

Cancel the common factor of $4 t + 3$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(5 t - 1) (4 t + 3)}{4 ((4 t + 3) (4 t - 3))}$

Step 3.3.2

Rewrite the expression.

$\frac{5 t - 1}{4 (4 t - 3)}$