Algebra Examples

$f (x) = - {(3 x^{2} + 1)}^{3}$

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 $x$ .

$0 = - {(3 x^{2} + 1)}^{3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- {(3 x^{2} + 1)}^{3} = 0$ .

$- {(3 x^{2} + 1)}^{3} = 0$

Step 1.2.2

Divide each term in $- {(3 x^{2} + 1)}^{3} = 0$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- {(3 x^{2} + 1)}^{3} = 0$ by $- 1$ .

$\frac{- {(3 x^{2} + 1)}^{3}}{- 1} = \frac{0}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{{(3 x^{2} + 1)}^{3}}{1} = \frac{0}{- 1}$

Step 1.2.2.2.2

Divide ${(3 x^{2} + 1)}^{3}$ by $1$ .

${(3 x^{2} + 1)}^{3} = \frac{0}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 1$ .

${(3 x^{2} + 1)}^{3} = 0$

Step 1.2.3

Set the $3 x^{2} + 1$ equal to $0$ .

$3 x^{2} + 1 = 0$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Subtract $1$ from both sides of the equation.

$3 x^{2} = - 1$

Step 1.2.4.2

Divide each term in $3 x^{2} = - 1$ by $3$ and simplify.

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Step 1.2.4.2.1

Divide each term in $3 x^{2} = - 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Step 1.2.4.2.2.1.2

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

$x^{2} = \frac{- 1}{3}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{3}$

Step 1.2.4.3

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

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

Step 1.2.4.4

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

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Step 1.2.4.4.1

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

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Step 1.2.4.4.1.1

Rewrite $- 1$ as $i^{2}$ .

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

Step 1.2.4.4.1.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.4.4.2

Pull terms out from under the radical.

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

Step 1.2.4.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{3}}$

Step 1.2.4.4.4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{3}}$

Step 1.2.4.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{3}}$

Step 1.2.4.4.6

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.2.4.4.7

Combine and simplify the denominator.

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Step 1.2.4.4.7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.2.4.4.7.2

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

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

Step 1.2.4.4.7.3

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

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

Step 1.2.4.4.7.4

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

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

Step 1.2.4.4.7.5

Add $1$ and $1$ .

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

Step 1.2.4.4.7.6

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

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Step 1.2.4.4.7.6.1

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

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

Step 1.2.4.4.7.6.2

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

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

Step 1.2.4.4.7.6.3

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

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

Step 1.2.4.4.7.6.4

Cancel the common factor of $2$ .

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Step 1.2.4.4.7.6.4.1

Cancel the common factor.

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

Step 1.2.4.4.7.6.4.2

Rewrite the expression.

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

Step 1.2.4.4.7.6.5

Evaluate the exponent.

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

Step 1.2.4.4.8

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

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

Step 1.2.4.5

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

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Step 1.2.4.5.1

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

$x = \frac{i \sqrt{3}}{3}$

Step 1.2.4.5.2

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

$x = - \frac{i \sqrt{3}}{3}$

Step 1.2.4.5.3

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

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y = - {(3 {(0)}^{2} + 1)}^{3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - {(3 \cdot 0^{2} + 1)}^{3}$

Step 2.2.2

Remove parentheses.

$y = - {(3 {(0)}^{2} + 1)}^{3}$

Step 2.2.3

Simplify $- {(3 {(0)}^{2} + 1)}^{3}$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = - {(3 \cdot 0 + 1)}^{3}$

Step 2.2.3.1.2

Multiply $3$ by $0$ .

$y = - {(0 + 1)}^{3}$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Add $0$ and $1$ .

$y = - 1^{3}$

Step 2.2.3.2.2

One to any power is one.

$y = - 1 \cdot 1$

Step 2.2.3.2.3

Multiply $- 1$ by $1$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 1)$

Step 4