Precalculus Examples

$f (x) = 33 x^{4} - x^{2} + 121$

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 $x$ .

$0 = 33 x^{4} - x^{2} + 121$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $33 x^{4} - x^{2} + 121 = 0$ .

$33 x^{4} - x^{2} + 121 = 0$

Step 1.2.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$33 u^{2} - u + 121 = 0$

$u = x^{2}$

Step 1.2.3

Use the quadratic formula to find the solutions.

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

Step 1.2.4

Substitute the values $a = 33$ , $b = - 1$ , and $c = 121$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (33 \cdot 121)}}{2 \cdot 33}$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 33 \cdot 121}}{2 \cdot 33}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 33 \cdot 121$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $33$ .

$u = \frac{1 \pm \sqrt{1 - 132 \cdot 121}}{2 \cdot 33}$

Step 1.2.5.1.2.2

Multiply $- 132$ by $121$ .

$u = \frac{1 \pm \sqrt{1 - 15972}}{2 \cdot 33}$

Step 1.2.5.1.3

Subtract $15972$ from $1$ .

$u = \frac{1 \pm \sqrt{- 15971}}{2 \cdot 33}$

Step 1.2.5.1.4

Rewrite $- 15971$ as $- 1 (15971)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 15971}}{2 \cdot 33}$

Step 1.2.5.1.5

Rewrite $\sqrt{- 1 (15971)}$ as $\sqrt{- 1} \cdot \sqrt{15971}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{15971}}{2 \cdot 33}$

Step 1.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{1 \pm i \sqrt{15971}}{2 \cdot 33}$

Step 1.2.5.2

Multiply $2$ by $33$ .

$u = \frac{1 \pm i \sqrt{15971}}{66}$

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 33 \cdot 121}}{2 \cdot 33}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 33 \cdot 121$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $33$ .

$u = \frac{1 \pm \sqrt{1 - 132 \cdot 121}}{2 \cdot 33}$

Step 1.2.6.1.2.2

Multiply $- 132$ by $121$ .

$u = \frac{1 \pm \sqrt{1 - 15972}}{2 \cdot 33}$

Step 1.2.6.1.3

Subtract $15972$ from $1$ .

$u = \frac{1 \pm \sqrt{- 15971}}{2 \cdot 33}$

Step 1.2.6.1.4

Rewrite $- 15971$ as $- 1 (15971)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 15971}}{2 \cdot 33}$

Step 1.2.6.1.5

Rewrite $\sqrt{- 1 (15971)}$ as $\sqrt{- 1} \cdot \sqrt{15971}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{15971}}{2 \cdot 33}$

Step 1.2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{1 \pm i \sqrt{15971}}{2 \cdot 33}$

Step 1.2.6.2

Multiply $2$ by $33$ .

$u = \frac{1 \pm i \sqrt{15971}}{66}$

Step 1.2.6.3

Change the $\pm$ to $+$ .

$u = \frac{1 + i \sqrt{15971}}{66}$

Step 1.2.7

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

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 33 \cdot 121}}{2 \cdot 33}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 33 \cdot 121$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $33$ .

$u = \frac{1 \pm \sqrt{1 - 132 \cdot 121}}{2 \cdot 33}$

Step 1.2.7.1.2.2

Multiply $- 132$ by $121$ .

$u = \frac{1 \pm \sqrt{1 - 15972}}{2 \cdot 33}$

Step 1.2.7.1.3

Subtract $15972$ from $1$ .

$u = \frac{1 \pm \sqrt{- 15971}}{2 \cdot 33}$

Step 1.2.7.1.4

Rewrite $- 15971$ as $- 1 (15971)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 15971}}{2 \cdot 33}$

Step 1.2.7.1.5

Rewrite $\sqrt{- 1 (15971)}$ as $\sqrt{- 1} \cdot \sqrt{15971}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{15971}}{2 \cdot 33}$

Step 1.2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{1 \pm i \sqrt{15971}}{2 \cdot 33}$

Step 1.2.7.2

Multiply $2$ by $33$ .

$u = \frac{1 \pm i \sqrt{15971}}{66}$

Step 1.2.7.3

Change the $\pm$ to $-$ .

$u = \frac{1 - i \sqrt{15971}}{66}$

Step 1.2.8

The final answer is the combination of both solutions.

$u = \frac{1 + i \sqrt{15971}}{66}, \frac{1 - i \sqrt{15971}}{66}$

Step 1.2.9

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = \frac{1 + i \sqrt{15971}}{66}$

${(x^{2})}^{1} = \frac{1 - i \sqrt{15971}}{66}$

Step 1.2.10

Solve the first equation for $x$ .

$x^{2} = \frac{1 + i \sqrt{15971}}{66}$

Step 1.2.11

Solve the equation for $x$ .

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Step 1.2.11.1

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

$x = \pm \sqrt{\frac{1 + i \sqrt{15971}}{66}}$

Step 1.2.11.2

Simplify $\pm \sqrt{\frac{1 + i \sqrt{15971}}{66}}$ .

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Step 1.2.11.2.1

Rewrite $\sqrt{\frac{1 + i \sqrt{15971}}{66}}$ as $\frac{\sqrt{1 + i \sqrt{15971}}}{\sqrt{66}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}}}{\sqrt{66}}$

Step 1.2.11.2.2

Multiply $\frac{\sqrt{1 + i \sqrt{15971}}}{\sqrt{66}}$ by $\frac{\sqrt{66}}{\sqrt{66}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}}}{\sqrt{66}} \cdot \frac{\sqrt{66}}{\sqrt{66}}$

Step 1.2.11.2.3

Combine and simplify the denominator.

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Step 1.2.11.2.3.1

Multiply $\frac{\sqrt{1 + i \sqrt{15971}}}{\sqrt{66}}$ by $\frac{\sqrt{66}}{\sqrt{66}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{\sqrt{66} \sqrt{66}}$

Step 1.2.11.2.3.2

Raise $\sqrt{66}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{1} \sqrt{66}}$

Step 1.2.11.2.3.3

Raise $\sqrt{66}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{1} {\sqrt{66}}^{1}}$

Step 1.2.11.2.3.4

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

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{1 + 1}}$

Step 1.2.11.2.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{2}}$

Step 1.2.11.2.3.6

Rewrite ${\sqrt{66}}^{2}$ as $66$ .

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Step 1.2.11.2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{66}$ as $66^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{{(66^{\frac{1}{2}})}^{2}}$

Step 1.2.11.2.3.6.2

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

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{66^{\frac{1}{2} \cdot 2}}$

Step 1.2.11.2.3.6.3

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

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{66^{\frac{2}{2}}}$

Step 1.2.11.2.3.6.4

Cancel the common factor of $2$ .

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Step 1.2.11.2.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{66^{\frac{2}{2}}}$

Step 1.2.11.2.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{66^{1}}$

Step 1.2.11.2.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 + i \sqrt{15971}} \sqrt{66}}{66}$

Step 1.2.11.2.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.11.3

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

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Step 1.2.11.3.1

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

$x = \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.11.3.2

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

$x = - \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.11.3.3

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

$x = \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}, - \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.12

Solve the second equation for $x$ .

${(x^{2})}^{1} = \frac{1 - i \sqrt{15971}}{66}$

Step 1.2.13

Solve the equation for $x$ .

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Step 1.2.13.1

Remove parentheses.

$x^{2} = \frac{1 - i \sqrt{15971}}{66}$

Step 1.2.13.2

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

$x = \pm \sqrt{\frac{1 - i \sqrt{15971}}{66}}$

Step 1.2.13.3

Simplify $\pm \sqrt{\frac{1 - i \sqrt{15971}}{66}}$ .

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Step 1.2.13.3.1

Rewrite $\sqrt{\frac{1 - i \sqrt{15971}}{66}}$ as $\frac{\sqrt{1 - i \sqrt{15971}}}{\sqrt{66}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}}}{\sqrt{66}}$

Step 1.2.13.3.2

Multiply $\frac{\sqrt{1 - i \sqrt{15971}}}{\sqrt{66}}$ by $\frac{\sqrt{66}}{\sqrt{66}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}}}{\sqrt{66}} \cdot \frac{\sqrt{66}}{\sqrt{66}}$

Step 1.2.13.3.3

Combine and simplify the denominator.

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Step 1.2.13.3.3.1

Multiply $\frac{\sqrt{1 - i \sqrt{15971}}}{\sqrt{66}}$ by $\frac{\sqrt{66}}{\sqrt{66}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{\sqrt{66} \sqrt{66}}$

Step 1.2.13.3.3.2

Raise $\sqrt{66}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{1} \sqrt{66}}$

Step 1.2.13.3.3.3

Raise $\sqrt{66}$ to the power of $1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{1} {\sqrt{66}}^{1}}$

Step 1.2.13.3.3.4

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

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{1 + 1}}$

Step 1.2.13.3.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{{\sqrt{66}}^{2}}$

Step 1.2.13.3.3.6

Rewrite ${\sqrt{66}}^{2}$ as $66$ .

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Step 1.2.13.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{66}$ as $66^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{{(66^{\frac{1}{2}})}^{2}}$

Step 1.2.13.3.3.6.2

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

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{66^{\frac{1}{2} \cdot 2}}$

Step 1.2.13.3.3.6.3

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

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{66^{\frac{2}{2}}}$

Step 1.2.13.3.3.6.4

Cancel the common factor of $2$ .

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Step 1.2.13.3.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{66^{\frac{2}{2}}}$

Step 1.2.13.3.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{66^{1}}$

Step 1.2.13.3.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 - i \sqrt{15971}} \sqrt{66}}{66}$

Step 1.2.13.3.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.13.4

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

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Step 1.2.13.4.1

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

$x = \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.13.4.2

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

$x = - \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.13.4.3

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

$x = \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}, - \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}$

Step 1.2.14

The solution to $33 x^{4} - x^{2} + 121 = 0$ is $x = \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}, - \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}, \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}, - \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}$ .

$x = \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}, - \frac{\sqrt{(1 + i \sqrt{15971}) \cdot 66}}{66}, \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}, - \frac{\sqrt{(1 - i \sqrt{15971}) \cdot 66}}{66}$

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 $y$ .

$y = 33 {(0)}^{4} - {(0)}^{2} + 121$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 33 \cdot 0^{4} - {(0)}^{2} + 121$

Step 2.2.2

Remove parentheses.

$y = 33 \cdot 0^{4} - 0^{2} + 121$

Step 2.2.3

Remove parentheses.

$y = 33 {(0)}^{4} - {(0)}^{2} + 121$

Step 2.2.4

Simplify $33 {(0)}^{4} - {(0)}^{2} + 121$ .

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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 $0$ .

$y = 33 \cdot 0 - {(0)}^{2} + 121$

Step 2.2.4.1.2

Multiply $33$ by $0$ .

$y = 0 - {(0)}^{2} + 121$

Step 2.2.4.1.3

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

$y = 0 - 0 + 121$

Step 2.2.4.1.4

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 121$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 121$

Step 2.2.4.2.2

Add $0$ and $121$ .

$y = 121$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 121)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 121)$

Step 4