Precalculus Examples

$3 x^{2} - x^{3}$

Step 1

Write $3 x^{2} - x^{3}$ as an equation.

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

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 $x$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

$x^{2} = 0$

$3 - x = 0$

Step 2.2.4

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

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Step 2.2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.2.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.2.5

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

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Step 2.2.5.1

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

$3 - x = 0$

Step 2.2.5.2

Solve $3 - x = 0$ for $x$ .

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Step 2.2.5.2.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 2.2.5.2.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 2.2.5.2.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 2.2.5.2.2.2

Simplify the left side.

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Step 2.2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 2.2.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 2.2.5.2.2.3

Simplify the right side.

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Step 2.2.5.2.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 2.2.6

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

$x = 0, 3$

Step 2.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

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

Step 3.2.2

Remove parentheses.

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

Step 3.2.3

Remove parentheses.

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

Step 3.2.4

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

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

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

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

Step 3.2.4.1.2

Multiply $3$ by $0$ .

$y = 0 - {(0)}^{3}$

Step 3.2.4.1.3

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

$y = 0 - 0$

Step 3.2.4.1.4

Multiply $- 1$ by $0$ .

$y = 0 + 0$

Step 3.2.4.2

Add $0$ and $0$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

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

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

Step 5