Basic Math Examples

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

Step 1

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

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Step 1.1

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 1.2

Write the factored form using these integers.

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 2.2

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 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

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

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Step 3.1

Multiply $\frac{(b - 1) (b + 4)}{{(b - 2)}^{2}}$ by $\frac{b + 4}{b - 3}$ .

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

Step 3.2

Raise $b + 4$ to the power of $1$ .

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

Step 3.3

Raise $b + 4$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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