Solve by Factoring 2x^4-x^2-1=0

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

Step 1

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

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

Step 2

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

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

Step 3

Factor by grouping.

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 $b = - 1$ .

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

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

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

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

Apply the distributive property.

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

Multiply $u$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Step 4

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

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

Step 5

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

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

Step 6

Factor.

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 $b = 1$ .

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

Remove unnecessary parentheses.

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

Step 7

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

$2 x^{2} + 1 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 8

Set $2 x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $2 x^{2} + 1$ equal to $0$ .

$2 x^{2} + 1 = 0$

Solve $2 x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x^{2} = - 1$

Divide each term in $2 x^{2} = - 1$ by $2$ and simplify.

Divide each term in $2 x^{2} = - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

Pull terms out from under the radical.

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

One to any power is one.

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

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

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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

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

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

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

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

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 9

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 10

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 11

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

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

Solve by Factoring 2x^4-x^2-1=0

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