Basic Math Examples

$(5 k^{2} - 7 k - 6) (\frac{k^{2} - 5 k + 6}{5 k^{2} - 7 k - 6})$

Step 1

Factor $k^{2} - 5 k + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.2

Write the factored form using these integers.

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{5 k^{2} - 7 k - 6}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 5 \cdot - 6 = - 30$ and whose sum is $b = - 7$ .

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Step 2.1.1

Factor $- 7$ out of $- 7 k$ .

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{5 k^{2} - 7 (k) - 6}$

Step 2.1.2

Rewrite $- 7$ as $3$ plus $- 10$

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{5 k^{2} + (3 - 10) k - 6}$

Step 2.1.3

Apply the distributive property.

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{5 k^{2} + 3 k - 10 k - 6}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{(5 k^{2} + 3 k) - 10 k - 6}$

Step 2.2.2

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

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{k (5 k + 3) - 2 (5 k + 3)}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $5 k + 3$ .

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{(5 k + 3) (k - 2)}$

Step 3

Cancel the common factor of $k - 2$ .

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Step 3.1

Cancel the common factor.

$(5 k^{2} - 7 k - 6) \frac{(k - 3) (k - 2)}{(5 k + 3) (k - 2)}$

Step 3.2

Rewrite the expression.

$(5 k^{2} - 7 k - 6) \frac{k - 3}{5 k + 3}$

Step 4

Multiply $5 k^{2} - 7 k - 6$ by $\frac{k - 3}{5 k + 3}$ .

$\frac{(5 k^{2} - 7 k - 6) (k - 3)}{5 k + 3}$

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 $a \cdot c = 5 \cdot - 6 = - 30$ and whose sum is $b = - 7$ .

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Step 5.1.1

Factor $- 7$ out of $- 7 k$ .

$\frac{(5 k^{2} - 7 (k) - 6) (k - 3)}{5 k + 3}$

Step 5.1.2

Rewrite $- 7$ as $3$ plus $- 10$

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

Step 5.1.3

Apply the distributive property.

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

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.

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

Step 5.2.2

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

$\frac{(k (5 k + 3) - 2 (5 k + 3)) (k - 3)}{5 k + 3}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $5 k + 3$ .

$\frac{(5 k + 3) (k - 2) (k - 3)}{5 k + 3}$

Step 6

Cancel the common factor of $5 k + 3$ .

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Step 6.1

Cancel the common factor.

$\frac{(5 k + 3) (k - 2) (k - 3)}{5 k + 3}$

Step 6.2

Divide $(k - 2) (k - 3)$ by $1$ .

$(k - 2) (k - 3)$

Step 7

Expand $(k - 2) (k - 3)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$k (k - 3) - 2 (k - 3)$

Step 7.2

Apply the distributive property.

$k \cdot k + k \cdot - 3 - 2 (k - 3)$

Step 7.3

Apply the distributive property.

$k \cdot k + k \cdot - 3 - 2 k - 2 \cdot - 3$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $k$ by $k$ .

$k^{2} + k \cdot - 3 - 2 k - 2 \cdot - 3$

Step 8.1.2

Move $- 3$ to the left of $k$ .

$k^{2} - 3 \cdot k - 2 k - 2 \cdot - 3$

Step 8.1.3

Multiply $- 2$ by $- 3$ .

$k^{2} - 3 k - 2 k + 6$

Step 8.2

Subtract $2 k$ from $- 3 k$ .

$k^{2} - 5 k + 6$