Basic Math Examples

$\frac{9 z^{2} - 88 z + 63}{9 z^{2} + 74 z - 63} \div \frac{81 z^{2} - 126 z + 49}{9 z^{2} + 74 z - 63}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9 z^{2} - 88 z + 63}{9 z^{2} + 74 z - 63} \cdot \frac{9 z^{2} + 74 z - 63}{81 z^{2} - 126 z + 49}$

Step 2

Cancel the common factor of $9 z^{2} + 74 z - 63$ .

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Step 2.1

Cancel the common factor.

$\frac{9 z^{2} - 88 z + 63}{9 z^{2} + 74 z - 63} \cdot \frac{9 z^{2} + 74 z - 63}{81 z^{2} - 126 z + 49}$

Step 2.2

Rewrite the expression.

$(9 z^{2} - 88 z + 63) \frac{1}{81 z^{2} - 126 z + 49}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $81 z^{2}$ as ${(9 z)}^{2}$ .

$(9 z^{2} - 88 z + 63) \frac{1}{{(9 z)}^{2} - 126 z + 49}$

Step 3.2

Rewrite $49$ as $7^{2}$ .

$(9 z^{2} - 88 z + 63) \frac{1}{{(9 z)}^{2} - 126 z + 7^{2}}$

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.

$126 z = 2 \cdot (9 z) \cdot 7$

Step 3.4

Rewrite the polynomial.

$(9 z^{2} - 88 z + 63) \frac{1}{{(9 z)}^{2} - 2 \cdot (9 z) \cdot 7 + 7^{2}}$

Step 3.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 9 z$ and $b = 7$ .

$(9 z^{2} - 88 z + 63) \frac{1}{{(9 z - 7)}^{2}}$

Step 4

Multiply $9 z^{2} - 88 z + 63$ by $\frac{1}{{(9 z - 7)}^{2}}$ .

$\frac{9 z^{2} - 88 z + 63}{{(9 z - 7)}^{2}}$

Step 5

Factor by grouping.

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Step 5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot 63 = 567$ and whose sum is $b = - 88$ .

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Step 5.1.1

Factor $- 88$ out of $- 88 z$ .

$\frac{9 z^{2} - 88 (z) + 63}{{(9 z - 7)}^{2}}$

Step 5.1.2

Rewrite $- 88$ as $- 7$ plus $- 81$

$\frac{9 z^{2} + (- 7 - 81) z + 63}{{(9 z - 7)}^{2}}$

Step 5.1.3

Apply the distributive property.

$\frac{9 z^{2} - 7 z - 81 z + 63}{{(9 z - 7)}^{2}}$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$\frac{(9 z^{2} - 7 z) - 81 z + 63}{{(9 z - 7)}^{2}}$

Step 5.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{z (9 z - 7) - 9 (9 z - 7)}{{(9 z - 7)}^{2}}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $9 z - 7$ .

$\frac{(9 z - 7) (z - 9)}{{(9 z - 7)}^{2}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $9 z - 7$ out of ${(9 z - 7)}^{2}$ .

$\frac{(9 z - 7) (z - 9)}{(9 z - 7) (9 z - 7)}$

Step 6.2

Cancel the common factor.

$\frac{(9 z - 7) (z - 9)}{(9 z - 7) (9 z - 7)}$

Step 6.3

Rewrite the expression.

$\frac{z - 9}{9 z - 7}$