Algebra Examples

$4 w^{4} + 40 w^{2} - 44 = 0$

Step 1

Factor $4$ out of $4 w^{4} + 40 w^{2} - 44$ .

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Step 1.1

Factor $4$ out of $4 w^{4}$ .

$4 (w^{4}) + 40 w^{2} - 44 = 0$

Step 1.2

Factor $4$ out of $40 w^{2}$ .

$4 (w^{4}) + 4 (10 w^{2}) - 44 = 0$

Step 1.3

Factor $4$ out of $- 44$ .

$4 w^{4} + 4 (10 w^{2}) + 4 \cdot - 11 = 0$

Step 1.4

Factor $4$ out of $4 w^{4} + 4 (10 w^{2})$ .

$4 (w^{4} + 10 w^{2}) + 4 \cdot - 11 = 0$

Step 1.5

Factor $4$ out of $4 (w^{4} + 10 w^{2}) + 4 \cdot - 11$ .

$4 (w^{4} + 10 w^{2} - 11) = 0$

Step 2

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

$4 ({(w^{2})}^{2} + 10 w^{2} - 11) = 0$

Step 3

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

$4 (u^{2} + 10 u - 11) = 0$

Step 4

Factor $u^{2} + 10 u - 11$ using the AC method.

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Step 4.1

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 $- 11$ and whose sum is $10$ .

$- 1, 11$

Step 4.2

Write the factored form using these integers.

$4 ((u - 1) (u + 11)) = 0$

Step 5

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

$4 ((w^{2} - 1) (w^{2} + 11)) = 0$

Step 6

Rewrite $1$ as $1^{2}$ .

$4 ((w^{2} - 1^{2}) (w^{2} + 11)) = 0$

Step 7

Factor.

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Step 7.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = w$ and $b = 1$ .

$4 ((w + 1) (w - 1) (w^{2} + 11)) = 0$

Step 7.2

Remove unnecessary parentheses.

$4 (w + 1) (w - 1) (w^{2} + 11) = 0$

Step 8

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

$w + 1 = 0$

$w - 1 = 0$

$w^{2} + 11 = 0$

Step 9

Set $w + 1$ equal to $0$ and solve for $w$ .

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Step 9.1

Set $w + 1$ equal to $0$ .

$w + 1 = 0$

Step 9.2

Subtract $1$ from both sides of the equation.

$w = - 1$

Step 10

Set $w - 1$ equal to $0$ and solve for $w$ .

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Step 10.1

Set $w - 1$ equal to $0$ .

$w - 1 = 0$

Step 10.2

Add $1$ to both sides of the equation.

$w = 1$

Step 11

Set $w^{2} + 11$ equal to $0$ and solve for $w$ .

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Step 11.1

Set $w^{2} + 11$ equal to $0$ .

$w^{2} + 11 = 0$

Step 11.2

Solve $w^{2} + 11 = 0$ for $w$ .

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Step 11.2.1

Subtract $11$ from both sides of the equation.

$w^{2} = - 11$

Step 11.2.2

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

$w = \pm \sqrt{- 11}$

Step 11.2.3

Simplify $\pm \sqrt{- 11}$ .

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Step 11.2.3.1

Rewrite $- 11$ as $- 1 (11)$ .

$w = \pm \sqrt{- 1 (11)}$

Step 11.2.3.2

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$w = \pm \sqrt{- 1} \cdot \sqrt{11}$

Step 11.2.3.3

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

$w = \pm i \sqrt{11}$

Step 11.2.4

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

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Step 11.2.4.1

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

$w = i \sqrt{11}$

Step 11.2.4.2

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

$w = - i \sqrt{11}$

Step 11.2.4.3

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

$w = i \sqrt{11}, - i \sqrt{11}$

Step 12

The final solution is all the values that make $4 (w + 1) (w - 1) (w^{2} + 11) = 0$ true.

$w = - 1, 1, i \sqrt{11}, - i \sqrt{11}$