Algebra Examples

$f (x) = 2 x^{3} + x^{2} + 10 x + 5$

Step 1

Set $2 x^{3} + x^{2} + 10 x + 5$ equal to $0$ .

$2 x^{3} + x^{2} + 10 x + 5 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor out the greatest common factor from each group.

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Step 2.1.1.1

Group the first two terms and the last two terms.

$(2 x^{3} + x^{2}) + 10 x + 5 = 0$

Step 2.1.1.2

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

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

Step 2.1.2

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

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

Step 2.2

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

$2 x + 1 = 0$

$x^{2} + 5 = 0$

Step 2.3

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

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Step 2.3.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 2.3.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.4

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

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Step 2.4.1

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

$x^{2} + 5 = 0$

Step 2.4.2

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

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Step 2.4.2.1

Subtract $5$ from both sides of the equation.

$x^{2} = - 5$

Step 2.4.2.2

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

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

Step 2.4.2.3

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

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Step 2.4.2.3.1

Rewrite $- 5$ as $- 1 (5)$ .

$x = \pm \sqrt{- 1 (5)}$

Step 2.4.2.3.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

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

Step 2.4.2.3.3

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

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

Step 2.4.2.4

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

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Step 2.4.2.4.1

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

$x = i \sqrt{5}$

Step 2.4.2.4.2

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

$x = - i \sqrt{5}$

Step 2.4.2.4.3

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

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

Step 2.5

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

$x = - \frac{1}{2}, i \sqrt{5}, - i \sqrt{5}$

Step 3