Calculus Examples

$y = x^{2} - 81$

Step 1

Set $x^{2} - 81$ equal to $0$ .

$x^{2} - 81 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $81$ to both sides of the equation.

$x^{2} = 81$

Step 2.2

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

$x = \pm \sqrt{81}$

Step 2.3

Simplify $\pm \sqrt{81}$ .

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Step 2.3.1

Rewrite $81$ as $9^{2}$ .

$x = \pm \sqrt{9^{2}}$

Step 2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 9$

Step 2.4

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

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Step 2.4.1

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

$x = 9$

Step 2.4.2

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

$x = - 9$

Step 2.4.3

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

$x = 9, - 9$

Step 2.5

The multiplicity of a root is the number of times the root appears. For example, a factor of ${(x + 5)}^{3}$ would have a root at $x = - 5$ with multiplicity of $3$ .

$x = 9$ (Multiplicity of $1$ )

$x = - 9$ (Multiplicity of $1$ )

$x = 9$ (Multiplicity of $1$ )

$x = - 9$ (Multiplicity of $1$ )

Step 3

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