Basic Math Examples

$\frac{2 r^{2} - 3 s}{12} \div (4 r^{2} - 12 r s + 9 s^{2})$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{2 r^{2} - 3 s}{12}}{4 r^{2} - 12 r s + 9 s^{2}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 r^{2} - 3 s}{12} \cdot \frac{1}{4 r^{2} - 12 r s + 9 s^{2}}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $4 r^{2}$ as ${(2 r)}^{2}$ .

$\frac{2 r^{2} - 3 s}{12} \cdot \frac{1}{{(2 r)}^{2} - 12 r s + 9 s^{2}}$

Step 3.2

Rewrite $9 s^{2}$ as ${(3 s)}^{2}$ .

$\frac{2 r^{2} - 3 s}{12} \cdot \frac{1}{{(2 r)}^{2} - 12 r s + {(3 s)}^{2}}$

Step 3.3

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

$12 r s = 2 \cdot (2 r) \cdot (3 s)$

Step 3.4

Rewrite the polynomial.

$\frac{2 r^{2} - 3 s}{12} \cdot \frac{1}{{(2 r)}^{2} - 2 \cdot (2 r) \cdot (3 s) + {(3 s)}^{2}}$

Step 3.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 r$ and $b = 3 s$ .

$\frac{2 r^{2} - 3 s}{12} \cdot \frac{1}{{(2 r - 3 s)}^{2}}$

Step 4

Multiply $\frac{2 r^{2} - 3 s}{12}$ by $\frac{1}{{(2 r - 3 s)}^{2}}$ .

$\frac{2 r^{2} - 3 s}{12 {(2 r - 3 s)}^{2}}$