Pre-Algebra Examples

$r^{4} - 8 r^{2} = - 15$

Step 1

Add $15$ to both sides of the equation.

$r^{4} - 8 r^{2} + 15 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Rewrite $r^{4}$ as ${(r^{2})}^{2}$ .

${(r^{2})}^{2} - 8 r^{2} + 15 = 0$

Step 2.2

Let $u = r^{2}$ . Substitute $u$ for all occurrences of $r^{2}$ .

$u^{2} - 8 u + 15 = 0$

Step 2.3

Factor $u^{2} - 8 u + 15$ using the AC method.

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Step 2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 2.3.2

Write the factored form using these integers.

$(u - 5) (u - 3) = 0$

Step 2.4

Replace all occurrences of $u$ with $r^{2}$ .

$(r^{2} - 5) (r^{2} - 3) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r^{2} - 5 = 0$

$r^{2} - 3 = 0$

Step 4

Set $r^{2} - 5$ equal to $0$ and solve for $r$ .

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Step 4.1

Set $r^{2} - 5$ equal to $0$ .

$r^{2} - 5 = 0$

Step 4.2

Solve $r^{2} - 5 = 0$ for $r$ .

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Step 4.2.1

Add $5$ to both sides of the equation.

$r^{2} = 5$

Step 4.2.2

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

$r = \pm \sqrt{5}$

Step 4.2.3

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

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Step 4.2.3.1

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

$r = \sqrt{5}$

Step 4.2.3.2

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

$r = - \sqrt{5}$

Step 4.2.3.3

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

$r = \sqrt{5}, - \sqrt{5}$

Step 5

Set $r^{2} - 3$ equal to $0$ and solve for $r$ .

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Step 5.1

Set $r^{2} - 3$ equal to $0$ .

$r^{2} - 3 = 0$

Step 5.2

Solve $r^{2} - 3 = 0$ for $r$ .

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Step 5.2.1

Add $3$ to both sides of the equation.

$r^{2} = 3$

Step 5.2.2

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

$r = \pm \sqrt{3}$

Step 5.2.3

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

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Step 5.2.3.1

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

$r = \sqrt{3}$

Step 5.2.3.2

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

$r = - \sqrt{3}$

Step 5.2.3.3

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

$r = \sqrt{3}, - \sqrt{3}$

Step 6

The final solution is all the values that make $(r^{2} - 5) (r^{2} - 3) = 0$ true.

$r = \sqrt{5}, - \sqrt{5}, \sqrt{3}, - \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$r = \sqrt{5}, - \sqrt{5}, \sqrt{3}, - \sqrt{3}$

Decimal Form:

$r = 2.23606797 \dots, - 2.23606797 \dots, 1.73205080 \dots, - 1.73205080 \dots$