Algebra Examples

$\frac{x + 10}{2 x^{2} + 38 x + 180}$

Step 1

Factor $2 x^{2} + 38 x + 180$ .

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Step 1.1

Factor $2$ out of $2 x^{2} + 38 x + 180$ .

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Step 1.1.1

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

$\frac{x + 10}{2 (x^{2}) + 38 x + 180}$

Step 1.1.2

Factor $2$ out of $38 x$ .

$\frac{x + 10}{2 (x^{2}) + 2 (19 x) + 180}$

Step 1.1.3

Factor $2$ out of $180$ .

$\frac{x + 10}{2 x^{2} + 2 (19 x) + 2 \cdot 90}$

Step 1.1.4

Factor $2$ out of $2 x^{2} + 2 (19 x)$ .

$\frac{x + 10}{2 (x^{2} + 19 x) + 2 \cdot 90}$

Step 1.1.5

Factor $2$ out of $2 (x^{2} + 19 x) + 2 \cdot 90$ .

$\frac{x + 10}{2 (x^{2} + 19 x + 90)}$

Step 1.2

Factor.

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Step 1.2.1

Factor $x^{2} + 19 x + 90$ using the AC method.

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Step 1.2.1.1

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 $90$ and whose sum is $19$ .

$9, 10$

Step 1.2.1.2

Write the factored form using these integers.

$\frac{x + 10}{2 ((x + 9) (x + 10))}$

Step 1.2.2

Remove unnecessary parentheses.

$\frac{x + 10}{2 (x + 9) (x + 10)}$

Step 2

Cancel the common factor of $x + 10$ .

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Step 2.1

Cancel the common factor.

$\frac{x + 10}{2 (x + 9) (x + 10)}$

Step 2.2

Rewrite the expression.

$\frac{1}{2 (x + 9)}$

Step 3

To find the holes in the graph, look at the denominator factors that were cancelled.

$x + 10$

Step 4

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{1}{2 (x + 9)}$ .

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Step 4.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 4.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 4.3

Substitute $- 10$ for $x$ in $\frac{1}{2 (x + 9)}$ and simplify.

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Step 4.3.1

Substitute $- 10$ for $x$ to find the $y$ coordinate of the hole.

$\frac{1}{2 (- 10 + 9)}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Add $- 10$ and $9$ .

$\frac{1}{2 \cdot - 1}$

Step 4.3.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{- 2}$

Step 4.3.2.3

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 4.4

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(- 10, - \frac{1}{2})$

Step 5