Finite Math Examples

$y = \frac{1}{4} x^{- 2} + 2$

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 $x$ .

$0 = \frac{1}{4} x^{- 2} + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{4} x^{- 2} + 2 = 0$ .

$\frac{1}{4} x^{- 2} + 2 = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{4} \cdot \frac{1}{x^{2}} + 2 = 0$

Step 1.2.2.2

Combine.

$\frac{1 \cdot 1}{4 x^{2}} + 2 = 0$

Step 1.2.2.3

Multiply $1$ by $1$ .

$\frac{1}{4 x^{2}} + 2 = 0$

Step 1.2.3

Subtract $2$ from both sides of the equation.

$\frac{1}{4 x^{2}} = - 2$

Step 1.2.4

Find the LCD of the terms in the equation.

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Step 1.2.4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$4 x^{2}, 1$

Step 1.2.4.2

The LCM of one and any expression is the expression.

$4 x^{2}$

Step 1.2.5

Multiply each term in $\frac{1}{4 x^{2}} = - 2$ by $4 x^{2}$ to eliminate the fractions.

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Step 1.2.5.1

Multiply each term in $\frac{1}{4 x^{2}} = - 2$ by $4 x^{2}$ .

$\frac{1}{4 x^{2}} (4 x^{2}) = - 2 (4 x^{2})$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Rewrite using the commutative property of multiplication.

$4 \frac{1}{4 x^{2}} x^{2} = - 2 (4 x^{2})$

Step 1.2.5.2.2

Cancel the common factor of $4$ .

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Step 1.2.5.2.2.1

Factor $4$ out of $4 x^{2}$ .

$4 \frac{1}{4 (x^{2})} x^{2} = - 2 (4 x^{2})$

Step 1.2.5.2.2.2

Cancel the common factor.

$4 \frac{1}{4 x^{2}} x^{2} = - 2 (4 x^{2})$

Step 1.2.5.2.2.3

Rewrite the expression.

$\frac{1}{x^{2}} x^{2} = - 2 (4 x^{2})$

Step 1.2.5.2.3

Cancel the common factor of $x^{2}$ .

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Step 1.2.5.2.3.1

Cancel the common factor.

$\frac{1}{x^{2}} x^{2} = - 2 (4 x^{2})$

Step 1.2.5.2.3.2

Rewrite the expression.

$1 = - 2 (4 x^{2})$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Multiply $4$ by $- 2$ .

$1 = - 8 x^{2}$

Step 1.2.6

Solve the equation.

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Step 1.2.6.1

Rewrite the equation as $- 8 x^{2} = 1$ .

$- 8 x^{2} = 1$

Step 1.2.6.2

Divide each term in $- 8 x^{2} = 1$ by $- 8$ and simplify.

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Step 1.2.6.2.1

Divide each term in $- 8 x^{2} = 1$ by $- 8$ .

$\frac{- 8 x^{2}}{- 8} = \frac{1}{- 8}$

Step 1.2.6.2.2

Simplify the left side.

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Step 1.2.6.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.6.2.2.1.1

Cancel the common factor.

$\frac{- 8 x^{2}}{- 8} = \frac{1}{- 8}$

Step 1.2.6.2.2.1.2

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

$x^{2} = \frac{1}{- 8}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{8}$

Step 1.2.6.3

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

$x = \pm \sqrt{- \frac{1}{8}}$

Step 1.2.6.4

Simplify $\pm \sqrt{- \frac{1}{8}}$ .

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Step 1.2.6.4.1

Rewrite $- \frac{1}{8}$ as ${(\frac{i}{2})}^{2} \frac{1}{2}$ .

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Step 1.2.6.4.1.1

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

$x = \pm \sqrt{i^{2} \frac{1}{8}}$

Step 1.2.6.4.1.2

Factor the perfect power $1^{2}$ out of $1$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 1}{8}}$

Step 1.2.6.4.1.3

Factor the perfect power $2^{2}$ out of $8$ .

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

Step 1.2.6.4.1.4

Rearrange the fraction $\frac{1^{2} \cdot 1}{2^{2} \cdot 2}$ .

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

Step 1.2.6.4.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

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

Step 1.2.6.4.2

Pull terms out from under the radical.

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

Step 1.2.6.4.3

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 1.2.6.4.4

Any root of $1$ is $1$ .

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

Step 1.2.6.4.5

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.2.6.4.6

Combine and simplify the denominator.

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Step 1.2.6.4.6.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.2.6.4.6.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.2.6.4.6.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.2.6.4.6.4

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

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

Step 1.2.6.4.6.5

Add $1$ and $1$ .

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

Step 1.2.6.4.6.6

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

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Step 1.2.6.4.6.6.1

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

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

Step 1.2.6.4.6.6.2

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

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

Step 1.2.6.4.6.6.3

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

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

Step 1.2.6.4.6.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.6.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.6.6.4.2

Rewrite the expression.

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

Step 1.2.6.4.6.6.5

Evaluate the exponent.

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

Step 1.2.6.4.7

Multiply $\frac{i}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 1.2.6.4.7.1

Multiply $\frac{i}{2}$ by $\frac{\sqrt{2}}{2}$ .

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

Step 1.2.6.4.7.2

Multiply $2$ by $2$ .

$x = \pm \frac{i \sqrt{2}}{4}$

Step 1.2.6.5

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

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Step 1.2.6.5.1

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

$x = \frac{i \sqrt{2}}{4}$

Step 1.2.6.5.2

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

$x = - \frac{i \sqrt{2}}{4}$

Step 1.2.6.5.3

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

$x = \frac{i \sqrt{2}}{4}, - \frac{i \sqrt{2}}{4}$

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 $y$ .

$y = \frac{1}{4} \cdot {(0)}^{- 2} + 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{4} \cdot 0^{- 2} + 2$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{4} \cdot {(0)}^{- 2} + 2$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = \frac{1}{4} \cdot \frac{1}{0^{2}} + 2$

Step 2.2.3.1.2

Combine.

$y = \frac{1 \cdot 1}{4 \cdot 0^{2}} + 2$

Step 2.2.3.1.3

Multiply $1$ by $1$ .

$y = \frac{1}{4 \cdot 0^{2}} + 2$

Step 2.2.3.1.4

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

$y = \frac{1}{4 \cdot 0} + 2$

Step 2.2.3.1.5

Multiply $4$ by $0$ .

$y = \frac{1}{0} + 2$

Step 2.2.3.2

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4