Precalculus Examples

$25 x^{4} + 26 x^{3} + 51 x^{2} + 52 x + 2$

Step 1

Set $25 x^{4} + 26 x^{3} + 51 x^{2} + 52 x + 2$ equal to $0$ .

$25 x^{4} + 26 x^{3} + 51 x^{2} + 52 x + 2 = 0$

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

Regroup terms.

$26 x^{3} + 52 x + 25 x^{4} + 51 x^{2} + 2 = 0$

Step 2.1.2

Factor $26 x$ out of $26 x^{3} + 52 x$ .

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Step 2.1.2.1

Factor $26 x$ out of $26 x^{3}$ .

$26 x (x^{2}) + 52 x + 25 x^{4} + 51 x^{2} + 2 = 0$

Step 2.1.2.2

Factor $26 x$ out of $52 x$ .

$26 x (x^{2}) + 26 x (2) + 25 x^{4} + 51 x^{2} + 2 = 0$

Step 2.1.2.3

Factor $26 x$ out of $26 x (x^{2}) + 26 x (2)$ .

$26 x (x^{2} + 2) + 25 x^{4} + 51 x^{2} + 2 = 0$

Step 2.1.3

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

$26 x (x^{2} + 2) + 25 {(x^{2})}^{2} + 51 x^{2} + 2 = 0$

Step 2.1.4

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

$26 x (x^{2} + 2) + 25 u^{2} + 51 u + 2 = 0$

Step 2.1.5

Factor by grouping.

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Step 2.1.5.1

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 $a \cdot c = 25 \cdot 2 = 50$ and whose sum is $b = 51$ .

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Step 2.1.5.1.1

Factor $51$ out of $51 u$ .

$26 x (x^{2} + 2) + 25 u^{2} + 51 (u) + 2 = 0$

Step 2.1.5.1.2

Rewrite $51$ as $1$ plus $50$

$26 x (x^{2} + 2) + 25 u^{2} + (1 + 50) u + 2 = 0$

Step 2.1.5.1.3

Apply the distributive property.

$26 x (x^{2} + 2) + 25 u^{2} + 1 u + 50 u + 2 = 0$

Step 2.1.5.1.4

Multiply $u$ by $1$ .

$26 x (x^{2} + 2) + 25 u^{2} + u + 50 u + 2 = 0$

Step 2.1.5.2

Factor out the greatest common factor from each group.

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Step 2.1.5.2.1

Group the first two terms and the last two terms.

$26 x (x^{2} + 2) + 25 u^{2} + u + 50 u + 2 = 0$

Step 2.1.5.2.2

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

$26 x (x^{2} + 2) + u (25 u + 1) + 2 (25 u + 1) = 0$

Step 2.1.5.3

Factor the polynomial by factoring out the greatest common factor, $25 u + 1$ .

$26 x (x^{2} + 2) + (25 u + 1) (u + 2) = 0$

Step 2.1.6

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

$26 x (x^{2} + 2) + (25 x^{2} + 1) (x^{2} + 2) = 0$

Step 2.1.7

Factor $x^{2} + 2$ out of $26 x (x^{2} + 2) + (25 x^{2} + 1) (x^{2} + 2)$ .

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Step 2.1.7.1

Factor $x^{2} + 2$ out of $26 x (x^{2} + 2)$ .

$(x^{2} + 2) (26 x) + (25 x^{2} + 1) (x^{2} + 2) = 0$

Step 2.1.7.2

Factor $x^{2} + 2$ out of $(25 x^{2} + 1) (x^{2} + 2)$ .

$(x^{2} + 2) (26 x) + (x^{2} + 2) (25 x^{2} + 1) = 0$

Step 2.1.7.3

Factor $x^{2} + 2$ out of $(x^{2} + 2) (26 x) + (x^{2} + 2) (25 x^{2} + 1)$ .

$(x^{2} + 2) (26 x + 25 x^{2} + 1) = 0$

Step 2.1.8

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

$(x^{2} + 2) (26 u + 25 u^{2} + 1) = 0$

Step 2.1.9

Factor by grouping.

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Step 2.1.9.1

Reorder terms.

$(x^{2} + 2) (25 u^{2} + 26 u + 1) = 0$

Step 2.1.9.2

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 $a \cdot c = 25 \cdot 1 = 25$ and whose sum is $b = 26$ .

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Step 2.1.9.2.1

Factor $26$ out of $26 u$ .

$(x^{2} + 2) (25 u^{2} + 26 (u) + 1) = 0$

Step 2.1.9.2.2

Rewrite $26$ as $1$ plus $25$

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

Step 2.1.9.2.3

Apply the distributive property.

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

Step 2.1.9.2.4

Multiply $u$ by $1$ .

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

Step 2.1.9.3

Factor out the greatest common factor from each group.

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Step 2.1.9.3.1

Group the first two terms and the last two terms.

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

Step 2.1.9.3.2

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

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

Step 2.1.9.4

Factor the polynomial by factoring out the greatest common factor, $25 u + 1$ .

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

Step 2.1.10

Factor.

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Step 2.1.10.1

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

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

Step 2.1.10.2

Remove unnecessary parentheses.

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

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 $0$ .

$x^{2} + 2 = 0$

$25 x + 1 = 0$

$x + 1 = 0$

Step 2.3

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

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Step 2.3.1

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

$x^{2} + 2 = 0$

Step 2.3.2

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

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Step 2.3.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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Step 2.3.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 2.3.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 2.3.2.3.3

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

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

Step 2.3.2.4

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

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Step 2.3.2.4.1

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

$x = i \sqrt{2}$

Step 2.3.2.4.2

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

$x = - i \sqrt{2}$

Step 2.3.2.4.3

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

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

Step 2.4

Set $25 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $25 x + 1$ equal to $0$ .

$25 x + 1 = 0$

Step 2.4.2

Solve $25 x + 1 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $1$ from both sides of the equation.

$25 x = - 1$

Step 2.4.2.2

Divide each term in $25 x = - 1$ by $25$ and simplify.

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Step 2.4.2.2.1

Divide each term in $25 x = - 1$ by $25$ .

$\frac{25 x}{25} = \frac{- 1}{25}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $25$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{25 x}{25} = \frac{- 1}{25}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{25}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{25}$

Step 2.5

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

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Step 2.5.1

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

$x + 1 = 0$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.6

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

$x = i \sqrt{2}, - i \sqrt{2}, - \frac{1}{25}, - 1$

Step 3