Algebra Examples

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

Step 1

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

$x^{5} - 3 x^{3} + x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x$ out of $x^{5} - 3 x^{3} + x$ .

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Step 2.1.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 3 x^{3} + x = 0$

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Factor $x$ out of $x \cdot x^{4} + x (- 3 x^{2})$ .

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

Step 2.1.6

Factor $x$ out of $x (x^{4} - 3 x^{2}) + x \cdot 1$ .

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

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 $0$ .

$x = 0$

$x^{4} - 3 x^{2} + 1 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x^{4} - 3 x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{4} - 3 x^{2} + 1$ equal to $0$ .

$x^{4} - 3 x^{2} + 1 = 0$

Step 2.4.2

Solve $x^{4} - 3 x^{2} + 1 = 0$ for $x$ .

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Step 2.4.2.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

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

$u = x^{2}$

Step 2.4.2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.2.3

Substitute the values $a = 1$ , $b = - 3$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 2.4.2.4

Simplify.

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Step 2.4.2.4.1

Simplify the numerator.

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Step 2.4.2.4.1.1

Raise $- 3$ to the power of $2$ .

$u = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.4.1.2.2

Multiply $- 4$ by $1$ .

$u = \frac{3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Step 2.4.2.4.1.3

Subtract $4$ from $9$ .

$u = \frac{3 \pm \sqrt{5}}{2 \cdot 1}$

Step 2.4.2.4.2

Multiply $2$ by $1$ .

$u = \frac{3 \pm \sqrt{5}}{2}$

Step 2.4.2.5

The final answer is the combination of both solutions.

$u = \frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$

Step 2.4.2.6

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 2.61803398$

${(x^{2})}^{1} = 0.38196601$

Step 2.4.2.7

Solve the first equation for $x$ .

$x^{2} = 2.61803398$

Step 2.4.2.8

Solve the equation for $x$ .

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Step 2.4.2.8.1

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

$x = \pm \sqrt{2.61803398}$

Step 2.4.2.8.2

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

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Step 2.4.2.8.2.1

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

$x = \sqrt{2.61803398}$

Step 2.4.2.8.2.2

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

$x = - \sqrt{2.61803398}$

Step 2.4.2.8.2.3

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

$x = \sqrt{2.61803398}, - \sqrt{2.61803398}$

Step 2.4.2.9

Solve the second equation for $x$ .

${(x^{2})}^{1} = 0.38196601$

Step 2.4.2.10

Solve the equation for $x$ .

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Step 2.4.2.10.1

Remove parentheses.

$x^{2} = 0.38196601$

Step 2.4.2.10.2

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

$x = \pm \sqrt{0.38196601}$

Step 2.4.2.10.3

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

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Step 2.4.2.10.3.1

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

$x = \sqrt{0.38196601}$

Step 2.4.2.10.3.2

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

$x = - \sqrt{0.38196601}$

Step 2.4.2.10.3.3

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

$x = \sqrt{0.38196601}, - \sqrt{0.38196601}$

Step 2.4.2.11

The solution to $x^{4} - 3 x^{2} + 1 = 0$ is $x = \sqrt{2.61803398}, - \sqrt{2.61803398}, \sqrt{0.38196601}, - \sqrt{0.38196601}$ .

$x = \sqrt{2.61803398}, - \sqrt{2.61803398}, \sqrt{0.38196601}, - \sqrt{0.38196601}$

Step 2.5

The final solution is all the values that make $x (x^{4} - 3 x^{2} + 1) = 0$ true.

$x = 0, \sqrt{2.61803398}, - \sqrt{2.61803398}, \sqrt{0.38196601}, - \sqrt{0.38196601}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \sqrt{2.61803398}, - \sqrt{2.61803398}, \sqrt{0.38196601}, - \sqrt{0.38196601}$

Decimal Form:

$x = 0, 1.61803398 \dots, - 1.61803398 \dots, 0.61803398 \dots, - 0.61803398 \dots$

Step 4