Algebra Examples

$4 x^{3} + x^{2} + 100 x + 25 = 0$

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

$(4 x^{3} + x^{2}) + 100 x + 25 = 0$

Step 1.2

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

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

Step 2

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

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

Step 3

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

$4 x + 1 = 0$

$x^{2} + 25 = 0$

Step 4

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 4.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 4.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 4.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

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.

$x = - \frac{1}{4}$

Step 5

Set $x^{2} + 25$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 25$ equal to $0$ .

$x^{2} + 25 = 0$

Step 5.2

Solve $x^{2} + 25 = 0$ for $x$ .

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Step 5.2.1

Subtract $25$ from both sides of the equation.

$x^{2} = - 25$

Step 5.2.2

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

$x = \pm \sqrt{- 25}$

Step 5.2.3

Simplify $\pm \sqrt{- 25}$ .

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Step 5.2.3.1

Rewrite $- 25$ as $- 1 (25)$ .

$x = \pm \sqrt{- 1 (25)}$

Step 5.2.3.2

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

Step 5.2.3.3

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

$x = \pm i \cdot \sqrt{25}$

Step 5.2.3.4

Rewrite $25$ as $5^{2}$ .

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

Step 5.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 5$

Step 5.2.3.6

Move $5$ to the left of $i$ .

$x = \pm 5 i$

Step 5.2.4

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

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Step 5.2.4.1

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

$x = 5 i$

Step 5.2.4.2

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

$x = - 5 i$

Step 5.2.4.3

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

$x = 5 i, - 5 i$

Step 6

The final solution is all the values that make $(4 x + 1) (x^{2} + 25) = 0$ true.

$x = - \frac{1}{4}, 5 i, - 5 i$