Algebra Examples

$2 x^{4} - x^{2} - 1 = 0$

Step 1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$2 u^{2} - u - 1 = 0$

$u = x^{2}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 2.1.1

Factor $- 1$ out of $- u$ .

$2 u^{2} - (u) - 1 = 0$

Step 2.1.2

Rewrite $- 1$ as $1$ plus $- 2$

$2 u^{2} + (1 - 2) u - 1 = 0$

Step 2.1.3

Apply the distributive property.

$2 u^{2} + 1 u - 2 u - 1 = 0$

Step 2.1.4

Multiply $u$ by $1$ .

$2 u^{2} + u - 2 u - 1 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(2 u^{2} + u) - 2 u - 1 = 0$

Step 2.2.2

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

$u (2 u + 1) - (2 u + 1) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$(2 u + 1) (u - 1) = 0$

Step 3

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

$2 u + 1 = 0$

$u - 1 = 0$

Step 4

Set $2 u + 1$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $2 u + 1$ equal to $0$ .

$2 u + 1 = 0$

Step 4.2

Solve $2 u + 1 = 0$ for $u$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$2 u = - 1$

Step 4.2.2

Divide each term in $2 u = - 1$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 u = - 1$ by $2$ .

$\frac{2 u}{2} = \frac{- 1}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 1}{2}$

Step 4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{2}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{2}$

Step 5

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

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Step 5.1

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

$u - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$u = 1$

Step 6

The final solution is all the values that make $(2 u + 1) (u - 1) = 0$ true.

$u = - \frac{1}{2}, 1$

Step 7

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = - \frac{1}{2}$

${(x^{2})}^{1} = 1$

Step 8

Solve the first equation for $x$ .

$x^{2} = - \frac{1}{2}$

Step 9

Solve the equation for $x$ .

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Step 9.1

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

$x = \pm \sqrt{- \frac{1}{2}}$

Step 9.2

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

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Step 9.2.1

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

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Step 9.2.1.1

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

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

Step 9.2.1.2

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

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

Step 9.2.2

Pull terms out from under the radical.

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

Step 9.2.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{2}}$

Step 9.2.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{2}}$

Step 9.2.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{2}}$

Step 9.2.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 9.2.7

Combine and simplify the denominator.

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Step 9.2.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.2.7.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 9.2.7.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 9.2.7.4

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

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 9.2.7.5

Add $1$ and $1$ .

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

Step 9.2.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 9.2.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 9.2.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 9.2.7.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 9.2.7.6.4

Cancel the common factor of $2$ .

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Step 9.2.7.6.4.1

Cancel the common factor.

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

Step 9.2.7.6.4.2

Rewrite the expression.

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

Step 9.2.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2}}{2}$

Step 9.2.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$x = \pm \frac{i \sqrt{2}}{2}$

Step 9.3

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

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Step 9.3.1

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

$x = \frac{i \sqrt{2}}{2}$

Step 9.3.2

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

$x = - \frac{i \sqrt{2}}{2}$

Step 9.3.3

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

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}$

Step 10

Solve the second equation for $x$ .

${(x^{2})}^{1} = 1$

Step 11

Solve the equation for $x$ .

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Step 11.1

Remove parentheses.

$x^{2} = 1$

Step 11.2

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

$x = \pm \sqrt{1}$

Step 11.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 11.4

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

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Step 11.4.1

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

$x = 1$

Step 11.4.2

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

$x = - 1$

Step 11.4.3

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

$x = 1, - 1$

Step 12

The solution to $2 x^{4} - x^{2} - 1 = 0$ is $x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}, 1, - 1$ .

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}, 1, - 1$