Solve by Factoring 42x^4-x^2-1=0

$42 x^{4} - x^{2} - 1 = 0$

Step 1

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

$42 {(x^{2})}^{2} - x^{2} - 1 = 0$

Step 2

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

$42 u^{2} - u - 1 = 0$

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 $a \cdot c = 42 \cdot - 1 = - 42$ and whose sum is $b = - 1$ .

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

$42 u^{2} - (u) - 1 = 0$

Rewrite $- 1$ as $6$ plus $- 7$

$42 u^{2} + (6 - 7) u - 1 = 0$

Apply the distributive property.

$42 u^{2} + 6 u - 7 u - 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(42 u^{2} + 6 u) - 7 u - 1 = 0$

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

$6 u (7 u + 1) - (7 u + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $7 u + 1$ .

$(7 u + 1) (6 u - 1) = 0$

Step 4

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

$(7 x^{2} + 1) (6 x^{2} - 1) = 0$

Step 5

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

$7 x^{2} + 1 = 0$

$6 x^{2} - 1 = 0$

Step 6

Set $7 x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $7 x^{2} + 1$ equal to $0$ .

$7 x^{2} + 1 = 0$

Solve $7 x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$7 x^{2} = - 1$

Divide each term in $7 x^{2} = - 1$ by $7$ and simplify.

Divide each term in $7 x^{2} = - 1$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{- 1}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{- 1}{7}$

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

$x^{2} = \frac{- 1}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{7}$

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

$x = \pm \sqrt{- \frac{1}{7}}$

Simplify $\pm \sqrt{- \frac{1}{7}}$ .

Rewrite $- \frac{1}{7}$ as $i^{2} \frac{1^{2}}{7}$ .

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

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

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

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

Pull terms out from under the radical.

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

One to any power is one.

$x = \pm i \sqrt{\frac{1}{7}}$

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{7}}$

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{7}}$

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

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

Raise $\sqrt{7}$ to the power of $1$ .

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

Raise $\sqrt{7}$ to the power of $1$ .

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

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

$x = \pm i \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

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.

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

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

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

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

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

Step 7

Set $6 x^{2} - 1$ equal to $0$ and solve for $x$ .

Set $6 x^{2} - 1$ equal to $0$ .

$6 x^{2} - 1 = 0$

Solve $6 x^{2} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$6 x^{2} = 1$

Divide each term in $6 x^{2} = 1$ by $6$ and simplify.

Divide each term in $6 x^{2} = 1$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{1}{6}$

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

$x^{2} = \frac{1}{6}$

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

$x = \pm \sqrt{\frac{1}{6}}$

Simplify $\pm \sqrt{\frac{1}{6}}$ .

Rewrite $\sqrt{\frac{1}{6}}$ as $\frac{\sqrt{1}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{6}}$

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{6}}$

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{6}}{\sqrt{6} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

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

$x = \pm \frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{6}}{{\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{6}}{6^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{6}}{6}$

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.

$x = \frac{\sqrt{6}}{6}$

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

$x = - \frac{\sqrt{6}}{6}$

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

$x = \frac{\sqrt{6}}{6}, - \frac{\sqrt{6}}{6}$

Step 8

The final solution is all the values that make $(7 x^{2} + 1) (6 x^{2} - 1) = 0$ true.

$x = \frac{i \sqrt{7}}{7}, - \frac{i \sqrt{7}}{7}, \frac{\sqrt{6}}{6}, - \frac{\sqrt{6}}{6}$

Solve by Factoring 42x^4-x^2-1=0

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