Basic Math Examples

$\frac{q^{2}}{6 q^{2} - 13 q + 5} \cdot (6 q^{2} - 13 q + 5)$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 6 \cdot 5 = 30$ and whose sum is $b = - 13$ .

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Step 1.1.1

Factor $- 13$ out of $- 13 q$ .

$\frac{q^{2}}{6 q^{2} - 13 (q) + 5} \cdot (6 q^{2} - 13 q + 5)$

Step 1.1.2

Rewrite $- 13$ as $- 3$ plus $- 10$

$\frac{q^{2}}{6 q^{2} + (- 3 - 10) q + 5} \cdot (6 q^{2} - 13 q + 5)$

Step 1.1.3

Apply the distributive property.

$\frac{q^{2}}{6 q^{2} - 3 q - 10 q + 5} \cdot (6 q^{2} - 13 q + 5)$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{q^{2}}{(6 q^{2} - 3 q) - 10 q + 5} \cdot (6 q^{2} - 13 q + 5)$

Step 1.2.2

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

$\frac{q^{2}}{3 q (2 q - 1) - 5 (2 q - 1)} \cdot (6 q^{2} - 13 q + 5)$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $2 q - 1$ .

$\frac{q^{2}}{(2 q - 1) (3 q - 5)} \cdot (6 q^{2} - 13 q + 5)$

Step 2

Multiply $\frac{q^{2}}{(2 q - 1) (3 q - 5)}$ by $6 q^{2} - 13 q + 5$ .

$\frac{q^{2} (6 q^{2} - 13 q + 5)}{(2 q - 1) (3 q - 5)}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 6 \cdot 5 = 30$ and whose sum is $b = - 13$ .

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Step 3.1.1

Factor $- 13$ out of $- 13 q$ .

$\frac{q^{2} (6 q^{2} - 13 (q) + 5)}{(2 q - 1) (3 q - 5)}$

Step 3.1.2

Rewrite $- 13$ as $- 3$ plus $- 10$

$\frac{q^{2} (6 q^{2} + (- 3 - 10) q + 5)}{(2 q - 1) (3 q - 5)}$

Step 3.1.3

Apply the distributive property.

$\frac{q^{2} (6 q^{2} - 3 q - 10 q + 5)}{(2 q - 1) (3 q - 5)}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{q^{2} ((6 q^{2} - 3 q) - 10 q + 5)}{(2 q - 1) (3 q - 5)}$

Step 3.2.2

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

$\frac{q^{2} (3 q (2 q - 1) - 5 (2 q - 1))}{(2 q - 1) (3 q - 5)}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 q - 1$ .

$\frac{q^{2} ((2 q - 1) (3 q - 5))}{(2 q - 1) (3 q - 5)}$

$\frac{q^{2} (2 q - 1) (3 q - 5)}{(2 q - 1) (3 q - 5)}$

Step 4

Cancel the common factor of $2 q - 1$ .

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Step 4.1

Cancel the common factor.

$\frac{q^{2} (2 q - 1) (3 q - 5)}{(2 q - 1) (3 q - 5)}$

Step 4.2

Rewrite the expression.

$\frac{q^{2} (3 q - 5)}{3 q - 5}$

Step 5

Cancel the common factor of $3 q - 5$ .

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Step 5.1

Cancel the common factor.

$\frac{q^{2} (3 q - 5)}{3 q - 5}$

Step 5.2

Divide $q^{2}$ by $1$ .

$q^{2}$