Algebra Examples

$f (x) = 2 - \frac{3}{x^{2}}$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Subtract $2$ from both sides of the equation.

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

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

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

$x^{2}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 1.2.4

Multiply each term in $- \frac{3}{x^{2}} = - 2$ by $x^{2}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $- \frac{3}{x^{2}} = - 2$ by $x^{2}$ .

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Move the leading negative in $- \frac{3}{x^{2}}$ into the numerator.

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

Step 1.2.4.2.1.2

Cancel the common factor.

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

Step 1.2.4.2.1.3

Rewrite the expression.

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

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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Step 1.2.5.2.1

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

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

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.2.1.2

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

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

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Dividing two negative values results in a positive value.

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

Step 1.2.5.3

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

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

Step 1.2.5.4

Simplify $\pm \sqrt{\frac{3}{2}}$ .

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Step 1.2.5.4.1

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

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

Step 1.2.5.4.2

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

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

Step 1.2.5.4.3

Combine and simplify the denominator.

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Step 1.2.5.4.3.1

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

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

Step 1.2.5.4.3.2

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

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

Step 1.2.5.4.3.3

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

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

Step 1.2.5.4.3.4

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

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

Step 1.2.5.4.3.5

Add $1$ and $1$ .

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

Step 1.2.5.4.3.6

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

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Step 1.2.5.4.3.6.1

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

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

Step 1.2.5.4.3.6.2

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

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

Step 1.2.5.4.3.6.3

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

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

Step 1.2.5.4.3.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.3.6.4.1

Cancel the common factor.

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

Step 1.2.5.4.3.6.4.2

Rewrite the expression.

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

Step 1.2.5.4.3.6.5

Evaluate the exponent.

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

Step 1.2.5.4.4

Simplify the numerator.

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Step 1.2.5.4.4.1

Combine using the product rule for radicals.

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

Step 1.2.5.4.4.2

Multiply $3$ by $2$ .

$x = \pm \frac{\sqrt{6}}{2}$

Step 1.2.5.5

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

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Step 1.2.5.5.1

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

$x = \frac{\sqrt{6}}{2}$

Step 1.2.5.5.2

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

$x = - \frac{\sqrt{6}}{2}$

Step 1.2.5.5.3

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

$x = \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\sqrt{6}}{2}, 0), (- \frac{\sqrt{6}}{2}, 0)$

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 $y$ .

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

Step 2.2

The equation has an undefined fraction.

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): $(\frac{\sqrt{6}}{2}, 0), (- \frac{\sqrt{6}}{2}, 0)$

y-intercept(s): $None$

Step 4