Algebra Examples

$\frac{10 x^{2} - 50 x - 140}{2 x^{3} - 26 x^{2} + 84 x}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $10$ out of $10 x^{2} - 50 x - 140$ .

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Step 1.1.1

Factor $10$ out of $10 x^{2}$ .

$\frac{10 (x^{2}) - 50 x - 140}{2 x^{3} - 26 x^{2} + 84 x}$

Step 1.1.2

Factor $10$ out of $- 50 x$ .

$\frac{10 (x^{2}) + 10 (- 5 x) - 140}{2 x^{3} - 26 x^{2} + 84 x}$

Step 1.1.3

Factor $10$ out of $- 140$ .

$\frac{10 x^{2} + 10 (- 5 x) + 10 \cdot - 14}{2 x^{3} - 26 x^{2} + 84 x}$

Step 1.1.4

Factor $10$ out of $10 x^{2} + 10 (- 5 x)$ .

$\frac{10 (x^{2} - 5 x) + 10 \cdot - 14}{2 x^{3} - 26 x^{2} + 84 x}$

Step 1.1.5

Factor $10$ out of $10 (x^{2} - 5 x) + 10 \cdot - 14$ .

$\frac{10 (x^{2} - 5 x - 14)}{2 x^{3} - 26 x^{2} + 84 x}$

Step 1.2

Factor $x^{2} - 5 x - 14$ using the AC method.

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Step 1.2.1

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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 1.2.2

Write the factored form using these integers.

$\frac{10 ((x - 7) (x + 2))}{2 x^{3} - 26 x^{2} + 84 x}$

$\frac{10 (x - 7) (x + 2)}{2 x^{3} - 26 x^{2} + 84 x}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2 x$ out of $2 x^{3} - 26 x^{2} + 84 x$ .

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Step 2.1.1

Factor $2 x$ out of $2 x^{3}$ .

$\frac{10 (x - 7) (x + 2)}{2 x (x^{2}) - 26 x^{2} + 84 x}$

Step 2.1.2

Factor $2 x$ out of $- 26 x^{2}$ .

$\frac{10 (x - 7) (x + 2)}{2 x (x^{2}) + 2 x (- 13 x) + 84 x}$

Step 2.1.3

Factor $2 x$ out of $84 x$ .

$\frac{10 (x - 7) (x + 2)}{2 x (x^{2}) + 2 x (- 13 x) + 2 x (42)}$

Step 2.1.4

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 13 x)$ .

$\frac{10 (x - 7) (x + 2)}{2 x (x^{2} - 13 x) + 2 x (42)}$

Step 2.1.5

Factor $2 x$ out of $2 x (x^{2} - 13 x) + 2 x (42)$ .

$\frac{10 (x - 7) (x + 2)}{2 x (x^{2} - 13 x + 42)}$

Step 2.2

Factor $x^{2} - 13 x + 42$ using the AC method.

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Step 2.2.1

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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Step 2.2.2

Write the factored form using these integers.

$\frac{10 (x - 7) (x + 2)}{2 x ((x - 7) (x - 6))}$

$\frac{10 (x - 7) (x + 2)}{2 x (x - 7) (x - 6)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $10$ and $2$ .

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Step 3.1.1

Factor $2$ out of $10 (x - 7) (x + 2)$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $2 x (x - 7) (x - 6)$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x - 7$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$\frac{5 (x + 2)}{(x) (x - 6)}$

$\frac{5 (x + 2)}{x (x - 6)}$