Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Add $1$ to both sides of the equation.

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

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 $\frac{2}{x^{2}} = 1$ by $x^{2}$ to eliminate the fractions.

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Step 1.2.4.1

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

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

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

Cancel the common factor.

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

Step 1.2.4.2.1.2

Rewrite the expression.

$2 = 1 x^{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

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

$2 = x^{2}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

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

$x^{2} = 2$

Step 1.2.5.2

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

$x = \pm \sqrt{2}$

Step 1.2.5.3

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

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Step 1.2.5.3.1

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

$x = \sqrt{2}$

Step 1.2.5.3.2

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

$x = - \sqrt{2}$

Step 1.2.5.3.3

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

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

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

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

y-intercept(s): $None$

Step 4