Calculus Examples

$y = \frac{2 x + 1}{x^{2} + x - 6}$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = \frac{2 x + 1}{x^{2} + x - 6}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$2 x + 1 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 1.2.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 1.2.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

$x = \frac{- 1}{2}$

$x = \frac{- 1}{2}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

$x = - \frac{1}{2}$

$x = - \frac{1}{2}$

$x = - \frac{1}{2}$

$x = - \frac{1}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{2}, 0)$

x-intercept(s): $(- \frac{1}{2}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = \frac{2 (0) + 1}{{(0)}^{2} + 0 - 6}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2 (0) + 1}{{(0)}^{2} + 0 - 6}$

Step 2.2.2

Remove parentheses.

$y = \frac{2 (0) + 1}{0^{2} + 0 - 6}$

Step 2.2.3

Remove parentheses.

$y = \frac{2 (0) + 1}{{(0)}^{2} + 0 - 6}$

Step 2.2.4

Simplify $\frac{2 (0) + 1}{{(0)}^{2} + 0 - 6}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

Multiply $2$ by $0$ .

$y = \frac{0 + 1}{0^{2} + 0 - 6}$

Step 2.2.4.1.2

Add $0$ and $1$ .

$y = \frac{1}{0^{2} + 0 - 6}$

$y = \frac{1}{0^{2} + 0 - 6}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{1}{0 + 0 - 6}$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = \frac{1}{0 - 6}$

Step 2.2.4.2.3

Subtract $6$ from $0$ .

$y = \frac{1}{- 6}$

$y = \frac{1}{- 6}$

Step 2.2.4.3

Move the negative in front of the fraction.

$y = - \frac{1}{6}$

$y = - \frac{1}{6}$

$y = - \frac{1}{6}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{1}{6})$

y-intercept(s): $(0, - \frac{1}{6})$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{2}, 0)$

y-intercept(s): $(0, - \frac{1}{6})$

Step 4

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$