Algebra Examples

$\frac{x^{2} + x - 12}{x - 7}$

Step 1

Write $\frac{x^{2} + x - 12}{x - 7}$ as an equation.

$y = \frac{x^{2} + x - 12}{x - 7}$

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 $x$ .

$0 = \frac{x^{2} + x - 12}{x - 7}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$x^{2} + x - 12 = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

Factor $x^{2} + x - 12$ using the AC method.

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Step 2.2.2.1.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 2.2.2.1.2

Write the factored form using these integers.

$(x - 3) (x + 4) = 0$

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 $0$ .

$x - 3 = 0$

$x + 4 = 0$

Step 2.2.2.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.2.2.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.2.2.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.2.2.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.2.2.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.2.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.2.2.5

The final solution is all the values that make $(x - 3) (x + 4) = 0$ true.

$x = 3, - 4$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 4, 0)$

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 $y$ .

$y = \frac{{(0)}^{2} + 0 - 12}{(0) - 7}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{{(0)}^{2} + 0 - 12}{(0) - 7}$

Step 3.2.2

Remove parentheses.

$y = \frac{0^{2} + 0 - 12}{(0) - 7}$

Step 3.2.3

Remove parentheses.

$y = \frac{0^{2} + 0 - 12}{0 - 7}$

Step 3.2.4

Remove parentheses.

$y = \frac{{(0)}^{2} + 0 - 12}{(0) - 7}$

Step 3.2.5

Simplify $\frac{{(0)}^{2} + 0 - 12}{(0) - 7}$ .

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Step 3.2.5.1

Simplify the numerator.

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Step 3.2.5.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0 + 0 - 12}{0 - 7}$

Step 3.2.5.1.2

Add $0$ and $0$ .

$y = \frac{0 - 12}{0 - 7}$

Step 3.2.5.1.3

Subtract $12$ from $0$ .

$y = \frac{- 12}{0 - 7}$

Step 3.2.5.2

Reduce the expression by cancelling the common factors.

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Step 3.2.5.2.1

Subtract $7$ from $0$ .

$y = \frac{- 12}{- 7}$

Step 3.2.5.2.2

Dividing two negative values results in a positive value.

$y = \frac{12}{7}$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{12}{7})$

Step 4

List the intersections.

x-intercept(s): $(3, 0), (- 4, 0)$

y-intercept(s): $(0, \frac{12}{7})$

Step 5