Algebra Examples

$f (x) = 9 x^{2} - 5 x$ , $g (x) = x^{2} - 3 x - 10$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{9 x^{2} - 5 x}{x^{2} - 3 x - 10}$

Step 1.2

Simplify.

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Step 1.2.1

Factor $x$ out of $9 x^{2} - 5 x$ .

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Step 1.2.1.1

Factor $x$ out of $9 x^{2}$ .

$\frac{x (9 x) - 5 x}{x^{2} - 3 x - 10}$

Step 1.2.1.2

Factor $x$ out of $- 5 x$ .

$\frac{x (9 x) + x \cdot - 5}{x^{2} - 3 x - 10}$

Step 1.2.1.3

Factor $x$ out of $x (9 x) + x \cdot - 5$ .

$\frac{x (9 x - 5)}{x^{2} - 3 x - 10}$

Step 1.2.2

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 1.2.2.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 1.2.2.2

Write the factored form using these integers.

$\frac{x (9 x - 5)}{(x - 5) (x + 2)}$

Step 2

Set the denominator in $\frac{x (9 x - 5)}{(x - 5) (x + 2)}$ equal to $0$ to find where the expression is undefined.

$(x - 5) (x + 2) = 0$

Step 3

Solve for $x$ .

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Step 3.1

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

$x - 5 = 0$

$x + 2 = 0$

Step 3.2

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.2.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 3.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.4

The final solution is all the values that make $(x - 5) (x + 2) = 0$ true.

$x = 5, - 2$

Step 4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 2) \cup (- 2, 5) \cup (5, \infty)$

Set-Builder Notation:

${x | x \neq - 2, 5}$

Step 5