Algebra Examples

$3 x^{4} - x^{2} + 6 = 0$

Step 1

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

$3 u^{2} - u + 6 = 0$

$u = x^{2}$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 3$ , $b = - 1$ , and $c = 6$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (3 \cdot 6)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 6$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$u = \frac{1 \pm \sqrt{1 - 12 \cdot 6}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $6$ .

$u = \frac{1 \pm \sqrt{1 - 72}}{2 \cdot 3}$

Step 4.1.3

Subtract $72$ from $1$ .

$u = \frac{1 \pm \sqrt{- 71}}{2 \cdot 3}$

Step 4.1.4

Rewrite $- 71$ as $- 1 (71)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 71}}{2 \cdot 3}$

Step 4.1.5

Rewrite $\sqrt{- 1 (71)}$ as $\sqrt{- 1} \cdot \sqrt{71}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{71}}{2 \cdot 3}$

Step 4.1.6

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

$u = \frac{1 \pm i \sqrt{71}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$u = \frac{1 \pm i \sqrt{71}}{6}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot 6$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$u = \frac{1 \pm \sqrt{1 - 12 \cdot 6}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $6$ .

$u = \frac{1 \pm \sqrt{1 - 72}}{2 \cdot 3}$

Step 5.1.3

Subtract $72$ from $1$ .

$u = \frac{1 \pm \sqrt{- 71}}{2 \cdot 3}$

Step 5.1.4

Rewrite $- 71$ as $- 1 (71)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 71}}{2 \cdot 3}$

Step 5.1.5

Rewrite $\sqrt{- 1 (71)}$ as $\sqrt{- 1} \cdot \sqrt{71}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{71}}{2 \cdot 3}$

Step 5.1.6

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

$u = \frac{1 \pm i \sqrt{71}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$u = \frac{1 \pm i \sqrt{71}}{6}$

Step 5.3

Change the $\pm$ to $+$ .

$u = \frac{1 + i \sqrt{71}}{6}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot 6$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$u = \frac{1 \pm \sqrt{1 - 12 \cdot 6}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $6$ .

$u = \frac{1 \pm \sqrt{1 - 72}}{2 \cdot 3}$

Step 6.1.3

Subtract $72$ from $1$ .

$u = \frac{1 \pm \sqrt{- 71}}{2 \cdot 3}$

Step 6.1.4

Rewrite $- 71$ as $- 1 (71)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 71}}{2 \cdot 3}$

Step 6.1.5

Rewrite $\sqrt{- 1 (71)}$ as $\sqrt{- 1} \cdot \sqrt{71}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{71}}{2 \cdot 3}$

Step 6.1.6

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

$u = \frac{1 \pm i \sqrt{71}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$u = \frac{1 \pm i \sqrt{71}}{6}$

Step 6.3

Change the $\pm$ to $-$ .

$u = \frac{1 - i \sqrt{71}}{6}$

Step 7

The final answer is the combination of both solutions.

$u = \frac{1 + i \sqrt{71}}{6}, \frac{1 - i \sqrt{71}}{6}$

Step 8

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

$x^{2} = \frac{1 + i \sqrt{71}}{6}$

${(x^{2})}^{1} = \frac{1 - i \sqrt{71}}{6}$

Step 9

Solve the first equation for $x$ .

$x^{2} = \frac{1 + i \sqrt{71}}{6}$

Step 10

Solve the equation for $x$ .

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Step 10.1

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

$x = \pm \sqrt{\frac{1 + i \sqrt{71}}{6}}$

Step 10.2

Simplify $\pm \sqrt{\frac{1 + i \sqrt{71}}{6}}$ .

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Step 10.2.1

Rewrite $\sqrt{\frac{1 + i \sqrt{71}}{6}}$ as $\frac{\sqrt{1 + i \sqrt{71}}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{71}}}{\sqrt{6}}$

Step 10.2.2

Multiply $\frac{\sqrt{1 + i \sqrt{71}}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{71}}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 10.2.3

Combine and simplify the denominator.

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Step 10.2.3.1

Multiply $\frac{\sqrt{1 + i \sqrt{71}}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 10.2.3.2

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

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 10.2.3.3

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

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 10.2.3.4

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

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 10.2.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 10.2.3.6

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

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Step 10.2.3.6.1

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

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 10.2.3.6.2

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

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 10.2.3.6.3

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

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 10.2.3.6.4

Cancel the common factor of $2$ .

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Step 10.2.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 10.2.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{6^{1}}$

Step 10.2.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 + i \sqrt{71}} \sqrt{6}}{6}$

Step 10.2.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}$

Step 10.3

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

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Step 10.3.1

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

$x = \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}$

Step 10.3.2

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

$x = - \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}$

Step 10.3.3

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

$x = \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}, - \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}$

Step 11

Solve the second equation for $x$ .

${(x^{2})}^{1} = \frac{1 - i \sqrt{71}}{6}$

Step 12

Solve the equation for $x$ .

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Step 12.1

Remove parentheses.

$x^{2} = \frac{1 - i \sqrt{71}}{6}$

Step 12.2

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

$x = \pm \sqrt{\frac{1 - i \sqrt{71}}{6}}$

Step 12.3

Simplify $\pm \sqrt{\frac{1 - i \sqrt{71}}{6}}$ .

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Step 12.3.1

Rewrite $\sqrt{\frac{1 - i \sqrt{71}}{6}}$ as $\frac{\sqrt{1 - i \sqrt{71}}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{71}}}{\sqrt{6}}$

Step 12.3.2

Multiply $\frac{\sqrt{1 - i \sqrt{71}}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{71}}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 12.3.3

Combine and simplify the denominator.

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Step 12.3.3.1

Multiply $\frac{\sqrt{1 - i \sqrt{71}}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 12.3.3.2

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

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 12.3.3.3

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

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 12.3.3.4

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

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 12.3.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 12.3.3.6

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

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Step 12.3.3.6.1

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

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 12.3.3.6.2

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

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 12.3.3.6.3

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

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 12.3.3.6.4

Cancel the common factor of $2$ .

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Step 12.3.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 12.3.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{6^{1}}$

Step 12.3.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 - i \sqrt{71}} \sqrt{6}}{6}$

Step 12.3.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}$

Step 12.4

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

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Step 12.4.1

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

$x = \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}$

Step 12.4.2

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

$x = - \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}$

Step 12.4.3

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

$x = \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}, - \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}$

Step 13

The solution to $3 x^{4} - x^{2} + 6 = 0$ is $x = \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}, - \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}, \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}, - \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}$ .

$x = \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}, - \frac{\sqrt{(1 + i \sqrt{71}) \cdot 6}}{6}, \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}, - \frac{\sqrt{(1 - i \sqrt{71}) \cdot 6}}{6}$