Algebra Examples

${(r^{2} + 2)}^{2} + (r^{2} + 2) - 6 = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify ${(r^{2} + 2)}^{2} + r^{2} + 2 - 6$ .

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Step 1.1.1

Remove parentheses.

${(r^{2} + 2)}^{2} + r^{2} + 2 - 6 = 0$

Step 1.1.2

Subtract $6$ from $2$ .

${(r^{2} + 2)}^{2} + r^{2} - 4 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

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

$(r^{2} + 2) (r^{2} + 2) + r^{2} - 4 = 0$

Step 2.2

Expand $(r^{2} + 2) (r^{2} + 2)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$r^{2} (r^{2} + 2) + 2 (r^{2} + 2) + r^{2} - 4 = 0$

Step 2.2.2

Apply the distributive property.

$r^{2} r^{2} + r^{2} \cdot 2 + 2 (r^{2} + 2) + r^{2} - 4 = 0$

Step 2.2.3

Apply the distributive property.

$r^{2} r^{2} + r^{2} \cdot 2 + 2 r^{2} + 2 \cdot 2 + r^{2} - 4 = 0$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $r^{2}$ by $r^{2}$ by adding the exponents.

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Step 2.3.1.1.1

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

$r^{2 + 2} + r^{2} \cdot 2 + 2 r^{2} + 2 \cdot 2 + r^{2} - 4 = 0$

Step 2.3.1.1.2

Add $2$ and $2$ .

$r^{4} + r^{2} \cdot 2 + 2 r^{2} + 2 \cdot 2 + r^{2} - 4 = 0$

Step 2.3.1.2

Move $2$ to the left of $r^{2}$ .

$r^{4} + 2 r^{2} + 2 r^{2} + 2 \cdot 2 + r^{2} - 4 = 0$

Step 2.3.1.3

Multiply $2$ by $2$ .

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

Step 2.3.2

Add $2 r^{2}$ and $2 r^{2}$ .

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

Step 2.4

Add $4 r^{2}$ and $r^{2}$ .

$r^{4} + 5 r^{2} + 4 - 4 = 0$

Step 2.5

Subtract $4$ from $4$ .

$r^{4} + 5 r^{2} + 0 = 0$

Step 2.6

Add $r^{4} + 5 r^{2}$ and $0$ .

$r^{4} + 5 r^{2} = 0$

Step 2.7

Factor $r^{2}$ out of $r^{4} + 5 r^{2}$ .

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Step 2.7.1

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

$r^{2} r^{2} + 5 r^{2} = 0$

Step 2.7.2

Factor $r^{2}$ out of $5 r^{2}$ .

$r^{2} r^{2} + r^{2} \cdot 5 = 0$

Step 2.7.3

Factor $r^{2}$ out of $r^{2} r^{2} + r^{2} \cdot 5$ .

$r^{2} (r^{2} + 5) = 0$

Step 3

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

$r^{2} = 0$

$r^{2} + 5 = 0$

Step 4

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

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Step 4.1

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

$r^{2} = 0$

Step 4.2

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

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Step 4.2.1

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

$r = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

Rewrite $0$ as $0^{2}$ .

$r = \pm \sqrt{0^{2}}$

Step 4.2.2.2

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

$r = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$r = 0$

Step 5

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

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Step 5.1

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

$r^{2} + 5 = 0$

Step 5.2

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

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Step 5.2.1

Subtract $5$ from both sides of the equation.

$r^{2} = - 5$

Step 5.2.2

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

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

Step 5.2.3

Simplify $\pm \sqrt{- 5}$ .

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Step 5.2.3.1

Rewrite $- 5$ as $- 1 (5)$ .

$r = \pm \sqrt{- 1 (5)}$

Step 5.2.3.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$r = \pm \sqrt{- 1} \cdot \sqrt{5}$

Step 5.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$r = \pm i \sqrt{5}$

Step 5.2.4

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

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Step 5.2.4.1

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

$r = i \sqrt{5}$

Step 5.2.4.2

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

$r = - i \sqrt{5}$

Step 5.2.4.3

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

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

Step 6

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

$r = 0, i \sqrt{5}, - i \sqrt{5}$