Calculus Examples

$4 y = 20 x^{3} - 3 x^{5}$

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 $x$ .

$4 (0) = 20 x^{3} - 3 x^{5}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $20 x^{3} - 3 x^{5} = 4 (0)$ .

$20 x^{3} - 3 x^{5} = 4 (0)$

Step 1.2.2

Multiply $4$ by $0$ .

$20 x^{3} - 3 x^{5} = 0$

Step 1.2.3

Factor $x^{3}$ out of $20 x^{3} - 3 x^{5}$ .

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Step 1.2.3.1

Factor $x^{3}$ out of $20 x^{3}$ .

$x^{3} \cdot 20 - 3 x^{5} = 0$

Step 1.2.3.2

Factor $x^{3}$ out of $- 3 x^{5}$ .

$x^{3} \cdot 20 + x^{3} (- 3 x^{2}) = 0$

Step 1.2.3.3

Factor $x^{3}$ out of $x^{3} \cdot 20 + x^{3} (- 3 x^{2})$ .

$x^{3} (20 - 3 x^{2}) = 0$

Step 1.2.4

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

$x^{3} = 0$

$20 - 3 x^{2} = 0$

Step 1.2.5

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

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Step 1.2.5.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 1.2.5.2

Solve $x^{3} = 0$ for $x$ .

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Step 1.2.5.2.1

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

$x = \sqrt[3]{0}$

Step 1.2.5.2.2

Simplify $\sqrt[3]{0}$ .

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Step 1.2.5.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.2.5.2.2.2

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

$x = 0$

Step 1.2.6

Set $20 - 3 x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $20 - 3 x^{2}$ equal to $0$ .

$20 - 3 x^{2} = 0$

Step 1.2.6.2

Solve $20 - 3 x^{2} = 0$ for $x$ .

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Step 1.2.6.2.1

Subtract $20$ from both sides of the equation.

$- 3 x^{2} = - 20$

Step 1.2.6.2.2

Divide each term in $- 3 x^{2} = - 20$ by $- 3$ and simplify.

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Step 1.2.6.2.2.1

Divide each term in $- 3 x^{2} = - 20$ by $- 3$ .

$\frac{- 3 x^{2}}{- 3} = \frac{- 20}{- 3}$

Step 1.2.6.2.2.2

Simplify the left side.

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Step 1.2.6.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.6.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 x^{2}}{- 3} = \frac{- 20}{- 3}$

Step 1.2.6.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 20}{- 3}$

Step 1.2.6.2.2.3

Simplify the right side.

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Step 1.2.6.2.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{20}{3}$

Step 1.2.6.2.3

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

$x = \pm \sqrt{\frac{20}{3}}$

Step 1.2.6.2.4

Simplify $\pm \sqrt{\frac{20}{3}}$ .

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Step 1.2.6.2.4.1

Rewrite $\sqrt{\frac{20}{3}}$ as $\frac{\sqrt{20}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{20}}{\sqrt{3}}$

Step 1.2.6.2.4.2

Simplify the numerator.

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Step 1.2.6.2.4.2.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 1.2.6.2.4.2.1.1

Factor $4$ out of $20$ .

$x = \pm \frac{\sqrt{4 (5)}}{\sqrt{3}}$

Step 1.2.6.2.4.2.1.2

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{\sqrt{2^{2} \cdot 5}}{\sqrt{3}}$

Step 1.2.6.2.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{2 \sqrt{5}}{\sqrt{3}}$

Step 1.2.6.2.4.3

Multiply $\frac{2 \sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{2 \sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.2.6.2.4.4

Combine and simplify the denominator.

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Step 1.2.6.2.4.4.1

Multiply $\frac{2 \sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.2.6.2.4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.2.6.2.4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.2.6.2.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.2.6.2.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.2.6.2.4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.2.6.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.2.6.2.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.2.6.2.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.6.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.2.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.6.2.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{3^{1}}$

Step 1.2.6.2.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{5} \sqrt{3}}{3}$

Step 1.2.6.2.4.5

Simplify the numerator.

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Step 1.2.6.2.4.5.1

Combine using the product rule for radicals.

$x = \pm \frac{2 \sqrt{3 \cdot 5}}{3}$

Step 1.2.6.2.4.5.2

Multiply $3$ by $5$ .

$x = \pm \frac{2 \sqrt{15}}{3}$

Step 1.2.6.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{2 \sqrt{15}}{3}$

Step 1.2.6.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{2 \sqrt{15}}{3}$

Step 1.2.6.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{2 \sqrt{15}}{3}, - \frac{2 \sqrt{15}}{3}$

Step 1.2.7

The final solution is all the values that make $x^{3} (20 - 3 x^{2}) = 0$ true.

$x = 0, \frac{2 \sqrt{15}}{3}, - \frac{2 \sqrt{15}}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{2 \sqrt{15}}{3}, 0), (- \frac{2 \sqrt{15}}{3}, 0)$

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 $y$ .

$4 y = 20 {(0)}^{3} - 3 {(0)}^{5}$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

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

$4 y = 20 \cdot 0 - 3 {(0)}^{5}$

Step 2.2.1.2

Multiply $20$ by $0$ .

$4 y = 0 - 3 {(0)}^{5}$

Step 2.2.1.3

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

$4 y = 0 - 3 \cdot 0$

Step 2.2.1.4

Multiply $- 3$ by $0$ .

$4 y = 0 + 0$

Step 2.2.1.5

Add $0$ and $0$ .

$4 y = 0$

Step 2.2.2

Divide each term in $4 y = 0$ by $4$ and simplify.

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Step 2.2.2.1

Divide each term in $4 y = 0$ by $4$ .

$\frac{4 y}{4} = \frac{0}{4}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{0}{4}$

Step 2.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{4}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $0$ by $4$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{2 \sqrt{15}}{3}, 0), (- \frac{2 \sqrt{15}}{3}, 0)$

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

Step 4