Precalculus Examples

$3 x^{4} + 3 x^{3} - 17 x^{2} + x - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$3 x^{3} + x + 3 x^{4} - 17 x^{2} - 6 = 0$

Step 1.2

Factor $x$ out of $3 x^{3} + x$ .

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Step 1.2.1

Factor $x$ out of $3 x^{3}$ .

$x (3 x^{2}) + x + 3 x^{4} - 17 x^{2} - 6 = 0$

Step 1.2.2

Raise $x$ to the power of $1$ .

$x (3 x^{2}) + x + 3 x^{4} - 17 x^{2} - 6 = 0$

Step 1.2.3

Factor $x$ out of $x^{1}$ .

$x (3 x^{2}) + x \cdot 1 + 3 x^{4} - 17 x^{2} - 6 = 0$

Step 1.2.4

Factor $x$ out of $x (3 x^{2}) + x \cdot 1$ .

$x (3 x^{2} + 1) + 3 x^{4} - 17 x^{2} - 6 = 0$

Step 1.3

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

$x (3 x^{2} + 1) + 3 {(x^{2})}^{2} - 17 x^{2} - 6 = 0$

Step 1.4

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

$x (3 x^{2} + 1) + 3 u^{2} - 17 u - 6 = 0$

Step 1.5

Factor by grouping.

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Step 1.5.1

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 $a \cdot c = 3 \cdot - 6 = - 18$ and whose sum is $b = - 17$ .

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Step 1.5.1.1

Factor $- 17$ out of $- 17 u$ .

$x (3 x^{2} + 1) + 3 u^{2} - 17 u - 6 = 0$

Step 1.5.1.2

Rewrite $- 17$ as $1$ plus $- 18$

$x (3 x^{2} + 1) + 3 u^{2} + (1 - 18) u - 6 = 0$

Step 1.5.1.3

Apply the distributive property.

$x (3 x^{2} + 1) + 3 u^{2} + 1 u - 18 u - 6 = 0$

Step 1.5.1.4

Multiply $u$ by $1$ .

$x (3 x^{2} + 1) + 3 u^{2} + u - 18 u - 6 = 0$

Step 1.5.2

Factor out the greatest common factor from each group.

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Step 1.5.2.1

Group the first two terms and the last two terms.

$x (3 x^{2} + 1) + 3 u^{2} + u - 18 u - 6 = 0$

Step 1.5.2.2

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

$x (3 x^{2} + 1) + u (3 u + 1) - 6 (3 u + 1) = 0$

Step 1.5.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$x (3 x^{2} + 1) + (3 u + 1) (u - 6) = 0$

Step 1.6

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

$x (3 x^{2} + 1) + (3 x^{2} + 1) (x^{2} - 6) = 0$

Step 1.7

Factor $3 x^{2} + 1$ out of $x (3 x^{2} + 1) + (3 x^{2} + 1) (x^{2} - 6)$ .

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Step 1.7.1

Factor $3 x^{2} + 1$ out of $x (3 x^{2} + 1)$ .

$(3 x^{2} + 1) x + (3 x^{2} + 1) (x^{2} - 6) = 0$

Step 1.7.2

Factor $3 x^{2} + 1$ out of $(3 x^{2} + 1) x + (3 x^{2} + 1) (x^{2} - 6)$ .

$(3 x^{2} + 1) (x + x^{2} - 6) = 0$

Step 1.8

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(3 x^{2} + 1) (u + u^{2} - 6) = 0$

Step 1.9

Factor $u + u^{2} - 6$ using the AC method.

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Step 1.9.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.9.2

Write the factored form using these integers.

$(3 x^{2} + 1) ((u - 2) (u + 3)) = 0$

Step 1.10

Factor.

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Step 1.10.1

Replace all occurrences of $u$ with $x$ .

$(3 x^{2} + 1) ((x - 2) (x + 3)) = 0$

Step 1.10.2

Remove unnecessary parentheses.

$(3 x^{2} + 1) (x - 2) (x + 3) = 0$

Step 2

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

$3 x^{2} + 1 = 0$

$x - 2 = 0$

$x + 3 = 0$

Step 3

Set $3 x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x^{2} + 1$ equal to $0$ .

$3 x^{2} + 1 = 0$

Step 3.2

Solve $3 x^{2} + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$3 x^{2} = - 1$

Step 3.2.2

Divide each term in $3 x^{2} = - 1$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x^{2} = - 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Step 3.2.2.2.1.2

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

$x^{2} = \frac{- 1}{3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{3}$

Step 3.2.3

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

$x = \pm \sqrt{- \frac{1}{3}}$

Step 3.2.4

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

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Step 3.2.4.1

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

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Step 3.2.4.1.1

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

$x = \pm \sqrt{i^{2} \frac{1}{3}}$

Step 3.2.4.1.2

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

$x = \pm \sqrt{i^{2} \frac{1^{2}}{3}}$

Step 3.2.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{3}}$

Step 3.2.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{3}}$

Step 3.2.4.4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{3}}$

Step 3.2.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{3}}$

Step 3.2.4.6

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 3.2.4.7

Combine and simplify the denominator.

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Step 3.2.4.7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.2.4.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.2.4.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.2.4.7.4

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

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.2.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.2.4.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.2.4.7.6.2

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

$x = \pm i \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.2.4.7.6.3

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

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.4.7.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{3}}{3^{1}}$

Step 3.2.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{3}}{3}$

Step 3.2.4.8

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$x = \pm \frac{i \sqrt{3}}{3}$

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.

$x = \frac{i \sqrt{3}}{3}$

Step 3.2.5.2

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

$x = - \frac{i \sqrt{3}}{3}$

Step 3.2.5.3

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

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

Step 4

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6

The final solution is all the values that make $(3 x^{2} + 1) (x - 2) (x + 3) = 0$ true.

$x = \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}, 2, - 3$