Basic Math Examples

$\frac{b^{2} + 3 b - 4}{b^{2} - 4 b + 4} \div \frac{b + 4}{b - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{b^{2} + 3 b - 4}{b^{2} - 4 b + 4} \cdot \frac{b - 3}{b + 4}$

Step 2

Factor $b^{2} + 3 b - 4$ using the AC method.

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Step 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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 2.2

Write the factored form using these integers.

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 b = 2 \cdot b \cdot 2$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = 2$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $b + 4$ .

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Step 4.1.1

Factor $b + 4$ out of $(b - 1) (b + 4)$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Multiply $\frac{b - 1}{{(b - 2)}^{2}}$ by $b - 3$ .

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