Calculus Examples

$y = x^{3} - 3 x^{2} - 105 x$

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 $x$ .

$0 = x^{3} - 3 x^{2} - 105 x$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{3} - 3 x^{2} - 105 x = 0$ .

$x^{3} - 3 x^{2} - 105 x = 0$

Step 1.2.2

Factor $x$ out of $x^{3} - 3 x^{2} - 105 x$ .

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Step 1.2.2.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 3 x^{2} - 105 x = 0$

Step 1.2.2.2

Factor $x$ out of $- 3 x^{2}$ .

$x \cdot x^{2} + x (- 3 x) - 105 x = 0$

Step 1.2.2.3

Factor $x$ out of $- 105 x$ .

$x \cdot x^{2} + x (- 3 x) + x \cdot - 105 = 0$

Step 1.2.2.4

Factor $x$ out of $x \cdot x^{2} + x (- 3 x)$ .

$x (x^{2} - 3 x) + x \cdot - 105 = 0$

Step 1.2.2.5

Factor $x$ out of $x (x^{2} - 3 x) + x \cdot - 105$ .

$x (x^{2} - 3 x - 105) = 0$

Step 1.2.3

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

$x = 0$

$x^{2} - 3 x - 105 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $x^{2} - 3 x - 105$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x^{2} - 3 x - 105$ equal to $0$ .

$x^{2} - 3 x - 105 = 0$

Step 1.2.5.2

Solve $x^{2} - 3 x - 105 = 0$ for $x$ .

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Step 1.2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.5.2.2

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 105$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 105)}}{2 \cdot 1}$

Step 1.2.5.2.3

Simplify.

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Step 1.2.5.2.3.1

Simplify the numerator.

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Step 1.2.5.2.3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 105}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 105$ .

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 105}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2.2

Multiply $- 4$ by $- 105$ .

$x = \frac{3 \pm \sqrt{9 + 420}}{2 \cdot 1}$

Step 1.2.5.2.3.1.3

Add $9$ and $420$ .

$x = \frac{3 \pm \sqrt{429}}{2 \cdot 1}$

Step 1.2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{429}}{2}$

Step 1.2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.5.2.4.1

Simplify the numerator.

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Step 1.2.5.2.4.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 105}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 105$ .

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 105}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2.2

Multiply $- 4$ by $- 105$ .

$x = \frac{3 \pm \sqrt{9 + 420}}{2 \cdot 1}$

Step 1.2.5.2.4.1.3

Add $9$ and $420$ .

$x = \frac{3 \pm \sqrt{429}}{2 \cdot 1}$

Step 1.2.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{429}}{2}$

Step 1.2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{429}}{2}$

Step 1.2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.5.2.5.1

Simplify the numerator.

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Step 1.2.5.2.5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 105}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 105$ .

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 105}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2.2

Multiply $- 4$ by $- 105$ .

$x = \frac{3 \pm \sqrt{9 + 420}}{2 \cdot 1}$

Step 1.2.5.2.5.1.3

Add $9$ and $420$ .

$x = \frac{3 \pm \sqrt{429}}{2 \cdot 1}$

Step 1.2.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{429}}{2}$

Step 1.2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{429}}{2}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{429}}{2}, \frac{3 - \sqrt{429}}{2}$

Step 1.2.6

The final solution is all the values that make $x (x^{2} - 3 x - 105) = 0$ true.

$x = 0, \frac{3 + \sqrt{429}}{2}, \frac{3 - \sqrt{429}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{3 + \sqrt{429}}{2}, 0), (\frac{3 - \sqrt{429}}{2}, 0)$

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 $y$ .

$y = {(0)}^{3} - 3 {(0)}^{2} - 105 \cdot 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{3} - 3 {(0)}^{2} - 105 \cdot 0$

Step 2.2.2

Remove parentheses.

$y = 0^{3} - 3 \cdot 0^{2} - 105 \cdot 0$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{3} - 3 {(0)}^{2} - 105 \cdot 0$

Step 2.2.4

Simplify ${(0)}^{3} - 3 {(0)}^{2} - 105 \cdot 0$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = 0 - 3 {(0)}^{2} - 105 \cdot 0$

Step 2.2.4.1.2

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

$y = 0 - 3 \cdot 0 - 105 \cdot 0$

Step 2.2.4.1.3

Multiply $- 3$ by $0$ .

$y = 0 + 0 - 105 \cdot 0$

Step 2.2.4.1.4

Multiply $- 105$ by $0$ .

$y = 0 + 0 + 0$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{3 + \sqrt{429}}{2}, 0), (\frac{3 - \sqrt{429}}{2}, 0)$

y-intercept(s): $(0, 0)$

Step 4