Precalculus Examples

$125 x^{6} - 25 x^{4} - 20 x^{2} + 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$(125 x^{6} - 25 x^{4}) - 20 x^{2} + 4 = 0$

Step 1.1.2

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

$25 x^{4} (5 x^{2} - 1) - 4 (5 x^{2} - 1) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $5 x^{2} - 1$ .

$(5 x^{2} - 1) (25 x^{4} - 4) = 0$

Step 1.3

Rewrite $25 x^{4}$ as ${(5 x^{2})}^{2}$ .

$(5 x^{2} - 1) ({(5 x^{2})}^{2} - 4) = 0$

Step 1.4

Rewrite $4$ as $2^{2}$ .

$(5 x^{2} - 1) ({(5 x^{2})}^{2} - 2^{2}) = 0$

Step 1.5

Factor.

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Step 1.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x^{2}$ and $b = 2$ .

$(5 x^{2} - 1) ((5 x^{2} + 2) (5 x^{2} - 2)) = 0$

Step 1.5.2

Remove unnecessary parentheses.

$(5 x^{2} - 1) (5 x^{2} + 2) (5 x^{2} - 2) = 0$

Step 2

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

$5 x^{2} - 1 = 0$

$5 x^{2} + 2 = 0$

$5 x^{2} - 2 = 0$

Step 3

Set $5 x^{2} - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $5 x^{2} - 1$ equal to $0$ .

$5 x^{2} - 1 = 0$

Step 3.2

Solve $5 x^{2} - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$5 x^{2} = 1$

Step 3.2.2

Divide each term in $5 x^{2} = 1$ by $5$ and simplify.

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Step 3.2.2.1

Divide each term in $5 x^{2} = 1$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{1}{5}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{1}{5}$

Step 3.2.2.2.1.2

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

$x^{2} = \frac{1}{5}$

Step 3.2.3

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

$x = \pm \sqrt{\frac{1}{5}}$

Step 3.2.4

Simplify $\pm \sqrt{\frac{1}{5}}$ .

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Step 3.2.4.1

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{5}}$

Step 3.2.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{5}}$

Step 3.2.4.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 3.2.4.4

Combine and simplify the denominator.

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Step 3.2.4.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.2.4.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.2.4.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.2.4.4.4

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

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.2.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.2.4.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.2.4.4.6.2

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

$x = \pm \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.2.4.4.6.3

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

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.2.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{5}}{5^{1}}$

Step 3.2.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{5}}{5}$

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.

$x = \frac{\sqrt{5}}{5}$

Step 3.2.5.2

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

$x = - \frac{\sqrt{5}}{5}$

Step 3.2.5.3

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

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}$

Step 4

Set $5 x^{2} + 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $5 x^{2} + 2$ equal to $0$ .

$5 x^{2} + 2 = 0$

Step 4.2

Solve $5 x^{2} + 2 = 0$ for $x$ .

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$5 x^{2} = - 2$

Step 4.2.2

Divide each term in $5 x^{2} = - 2$ by $5$ and simplify.

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Step 4.2.2.1

Divide each term in $5 x^{2} = - 2$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{- 2}{5}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{- 2}{5}$

Step 4.2.2.2.1.2

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

$x^{2} = \frac{- 2}{5}$

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.

$x^{2} = - \frac{2}{5}$

Step 4.2.3

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

$x = \pm \sqrt{- \frac{2}{5}}$

Step 4.2.4

Simplify $\pm \sqrt{- \frac{2}{5}}$ .

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Step 4.2.4.1

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

$x = \pm \sqrt{i^{2} \frac{2}{5}}$

Step 4.2.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{2}{5}}$

Step 4.2.4.3

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{5}}$

Step 4.2.4.4

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i (\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 4.2.4.5

Combine and simplify the denominator.

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Step 4.2.4.5.1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.2.4.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 4.2.4.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 4.2.4.5.4

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

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.2.4.5.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.2.4.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.2.4.5.6.2

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

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.2.4.5.6.3

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

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.2.4.5.6.4

Cancel the common factor of $2$ .

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Step 4.2.4.5.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.2.4.5.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5^{1}}$

Step 4.2.4.5.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2} \sqrt{5}}{5}$

Step 4.2.4.6

Simplify the numerator.

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Step 4.2.4.6.1

Combine using the product rule for radicals.

$x = \pm i \frac{\sqrt{2 \cdot 5}}{5}$

Step 4.2.4.6.2

Multiply $2$ by $5$ .

$x = \pm i \frac{\sqrt{10}}{5}$

Step 4.2.4.7

Combine $i$ and $\frac{\sqrt{10}}{5}$ .

$x = \pm \frac{i \sqrt{10}}{5}$

Step 4.2.5

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

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Step 4.2.5.1

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

$x = \frac{i \sqrt{10}}{5}$

Step 4.2.5.2

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

$x = - \frac{i \sqrt{10}}{5}$

Step 4.2.5.3

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

$x = \frac{i \sqrt{10}}{5}, - \frac{i \sqrt{10}}{5}$

Step 5

Set $5 x^{2} - 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $5 x^{2} - 2$ equal to $0$ .

$5 x^{2} - 2 = 0$

Step 5.2

Solve $5 x^{2} - 2 = 0$ for $x$ .

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Step 5.2.1

Add $2$ to both sides of the equation.

$5 x^{2} = 2$

Step 5.2.2

Divide each term in $5 x^{2} = 2$ by $5$ and simplify.

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Step 5.2.2.1

Divide each term in $5 x^{2} = 2$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{2}{5}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{2}{5}$

Step 5.2.2.2.1.2

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

$x^{2} = \frac{2}{5}$

Step 5.2.3

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

$x = \pm \sqrt{\frac{2}{5}}$

Step 5.2.4

Simplify $\pm \sqrt{\frac{2}{5}}$ .

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Step 5.2.4.1

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{5}}$

Step 5.2.4.2

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.2.4.3

Combine and simplify the denominator.

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Step 5.2.4.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.2.4.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 5.2.4.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 5.2.4.3.4

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

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 5.2.4.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 5.2.4.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 5.2.4.3.6.2

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

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 5.2.4.3.6.3

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

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.2.4.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5^{1}}$

Step 5.2.4.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2} \sqrt{5}}{5}$

Step 5.2.4.4

Simplify the numerator.

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Step 5.2.4.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{2 \cdot 5}}{5}$

Step 5.2.4.4.2

Multiply $2$ by $5$ .

$x = \pm \frac{\sqrt{10}}{5}$

Step 5.2.5

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

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Step 5.2.5.1

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

$x = \frac{\sqrt{10}}{5}$

Step 5.2.5.2

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

$x = - \frac{\sqrt{10}}{5}$

Step 5.2.5.3

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

$x = \frac{\sqrt{10}}{5}, - \frac{\sqrt{10}}{5}$

Step 6

The final solution is all the values that make $(5 x^{2} - 1) (5 x^{2} + 2) (5 x^{2} - 2) = 0$ true.

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}, \frac{i \sqrt{10}}{5}, - \frac{i \sqrt{10}}{5}, \frac{\sqrt{10}}{5}, - \frac{\sqrt{10}}{5}$

Step 7