Algebra Examples

$9 x^{4} - 2 x^{2} - 7 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

$9 {(x^{2})}^{2} - 2 x^{2} - 7 = 0$

Step 1.2

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

$9 u^{2} - 2 u - 7 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot - 7 = - 63$ and whose sum is $b = - 2$ .

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Step 1.3.1.1

Factor $- 2$ out of $- 2 u$ .

$9 u^{2} - 2 u - 7 = 0$

Step 1.3.1.2

Rewrite $- 2$ as $7$ plus $- 9$

$9 u^{2} + (7 - 9) u - 7 = 0$

Step 1.3.1.3

Apply the distributive property.

$9 u^{2} + 7 u - 9 u - 7 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$(9 u^{2} + 7 u) - 9 u - 7 = 0$

Step 1.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (9 u + 7) - (9 u + 7) = 0$

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $9 u + 7$ .

$(9 u + 7) (u - 1) = 0$

Step 1.4

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

$(9 x^{2} + 7) (x^{2} - 1) = 0$

Step 1.5

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

$(9 x^{2} + 7) (x^{2} - 1^{2}) = 0$

Step 1.6

Factor.

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Step 1.6.1

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

$(9 x^{2} + 7) ((x + 1) (x - 1)) = 0$

Step 1.6.2

Remove unnecessary parentheses.

$(9 x^{2} + 7) (x + 1) (x - 1) = 0$

Step 2

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

$9 x^{2} + 7 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 3

Set $9 x^{2} + 7$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $9 x^{2} + 7$ equal to $0$ .

$9 x^{2} + 7 = 0$

Step 3.2

Solve $9 x^{2} + 7 = 0$ for $x$ .

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Step 3.2.1

Subtract $7$ from both sides of the equation.

$9 x^{2} = - 7$

Step 3.2.2

Divide each term in $9 x^{2} = - 7$ by $9$ and simplify.

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Step 3.2.2.1

Divide each term in $9 x^{2} = - 7$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{- 7}{9}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{- 7}{9}$

Step 3.2.2.2.1.2

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

$x^{2} = \frac{- 7}{9}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{7}{9}$

Step 3.2.3

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

$x = \pm \sqrt{- \frac{7}{9}}$

Step 3.2.4

Simplify $\pm \sqrt{- \frac{7}{9}}$ .

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Step 3.2.4.1

Rewrite $- \frac{7}{9}$ as ${(\frac{i}{3})}^{2} \cdot 7$ .

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Step 3.2.4.1.1

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

$x = \pm \sqrt{i^{2} \frac{7}{9}}$

Step 3.2.4.1.2

Factor the perfect power $1^{2}$ out of $7$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 7}{9}}$

Step 3.2.4.1.3

Factor the perfect power $3^{2}$ out of $9$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 7}{3^{2} \cdot 1}}$

Step 3.2.4.1.4

Rearrange the fraction $\frac{1^{2} \cdot 7}{3^{2} \cdot 1}$ .

$x = \pm \sqrt{i^{2} ({(\frac{1}{3})}^{2} \cdot 7)}$

Step 3.2.4.1.5

Rewrite $i^{2} {(\frac{1}{3})}^{2}$ as ${(\frac{i}{3})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{3})}^{2} \cdot 7}$

Step 3.2.4.2

Pull terms out from under the radical.

$x = \pm \frac{i}{3} \sqrt{7}$

Step 3.2.4.3

Combine $\frac{i}{3}$ and $\sqrt{7}$ .

$x = \pm \frac{i \sqrt{7}}{3}$

Step 3.2.5

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

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Step 3.2.5.1

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

$x = \frac{i \sqrt{7}}{3}$

Step 3.2.5.2

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

$x = - \frac{i \sqrt{7}}{3}$

Step 3.2.5.3

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

$x = \frac{i \sqrt{7}}{3}, - \frac{i \sqrt{7}}{3}$

Step 4

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

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

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

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Step 5.1

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

$x - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 6

The final solution is all the values that make $(9 x^{2} + 7) (x + 1) (x - 1) = 0$ true.

$x = \frac{i \sqrt{7}}{3}, - \frac{i \sqrt{7}}{3}, - 1, 1$