Algebra Examples

$\frac{2 x^{2} + 30 x + 108}{x^{2} + 9 x}$

Step 1

Write $\frac{2 x^{2} + 30 x + 108}{x^{2} + 9 x}$ as an equation.

$y = \frac{2 x^{2} + 30 x + 108}{x^{2} + 9 x}$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{2 x^{2} + 30 x + 108}{x^{2} + 9 x}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$2 x^{2} + 30 x + 108 = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

Factor the left side of the equation.

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Step 2.2.2.1.1

Factor $2$ out of $2 x^{2} + 30 x + 108$ .

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Step 2.2.2.1.1.1

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

$2 (x^{2}) + 30 x + 108 = 0$

Step 2.2.2.1.1.2

Factor $2$ out of $30 x$ .

$2 (x^{2}) + 2 (15 x) + 108 = 0$

Step 2.2.2.1.1.3

Factor $2$ out of $108$ .

$2 x^{2} + 2 (15 x) + 2 \cdot 54 = 0$

Step 2.2.2.1.1.4

Factor $2$ out of $2 x^{2} + 2 (15 x)$ .

$2 (x^{2} + 15 x) + 2 \cdot 54 = 0$

Step 2.2.2.1.1.5

Factor $2$ out of $2 (x^{2} + 15 x) + 2 \cdot 54$ .

$2 (x^{2} + 15 x + 54) = 0$

Step 2.2.2.1.2

Factor.

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Step 2.2.2.1.2.1

Factor $x^{2} + 15 x + 54$ using the AC method.

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Step 2.2.2.1.2.1.1

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 $54$ and whose sum is $15$ .

$6, 9$

Step 2.2.2.1.2.1.2

Write the factored form using these integers.

$2 ((x + 6) (x + 9)) = 0$

Step 2.2.2.1.2.2

Remove unnecessary parentheses.

$2 (x + 6) (x + 9) = 0$

Step 2.2.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 6 = 0$

$x + 9 = 0$

Step 2.2.2.3

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 2.2.2.3.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 2.2.2.3.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 2.2.2.4

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.2.2.4.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 2.2.2.4.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.2.2.5

The final solution is all the values that make $2 (x + 6) (x + 9) = 0$ true.

$x = - 6, - 9$

Step 2.2.3

Exclude the solutions that do not make $0 = \frac{2 x^{2} + 30 x + 108}{x^{2} + 9 x}$ true.

$x = - 6$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 6, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{2 {(0)}^{2} + 30 (0) + 108}{{(0)}^{2} + 9 (0)}$

Step 3.2

The equation has an undefined fraction.

Undefined

Step 3.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 4

List the intersections.

x-intercept(s): $(- 6, 0)$

y-intercept(s): $None$

Step 5