Finite Math Examples

$f (x) = x^{4} - 8 x^{3} + 12 x^{2} + 24 x - 45$

Step 1

Set $x^{4} - 8 x^{3} + 12 x^{2} + 24 x - 45$ equal to $0$ .

$x^{4} - 8 x^{3} + 12 x^{2} + 24 x - 45 = 0$

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.

$- 8 x^{3} + 24 x + x^{4} + 12 x^{2} - 45 = 0$

Step 2.1.2

Factor $- 8 x$ out of $- 8 x^{3} + 24 x$ .

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Step 2.1.2.1

Factor $- 8 x$ out of $- 8 x^{3}$ .

$- 8 x \cdot x^{2} + 24 x + x^{4} + 12 x^{2} - 45 = 0$

Step 2.1.2.2

Factor $- 8 x$ out of $24 x$ .

$- 8 x \cdot x^{2} - 8 x \cdot - 3 + x^{4} + 12 x^{2} - 45 = 0$

Step 2.1.2.3

Factor $- 8 x$ out of $- 8 x (x^{2}) - 8 x (- 3)$ .

$- 8 x (x^{2} - 3) + x^{4} + 12 x^{2} - 45 = 0$

Step 2.1.3

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

$- 8 x (x^{2} - 3) + {(x^{2})}^{2} + 12 x^{2} - 45 = 0$

Step 2.1.4

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

$- 8 x (x^{2} - 3) + u^{2} + 12 u - 45 = 0$

Step 2.1.5

Factor $u^{2} + 12 u - 45$ using the AC method.

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Step 2.1.5.1

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 $- 45$ and whose sum is $12$ .

$- 3, 15$

Step 2.1.5.2

Write the factored form using these integers.

$- 8 x (x^{2} - 3) + (u - 3) (u + 15) = 0$

Step 2.1.6

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

$- 8 x (x^{2} - 3) + (x^{2} - 3) (x^{2} + 15) = 0$

Step 2.1.7

Factor $x^{2} - 3$ out of $- 8 x (x^{2} - 3) + (x^{2} - 3) (x^{2} + 15)$ .

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Step 2.1.7.1

Factor $x^{2} - 3$ out of $- 8 x (x^{2} - 3)$ .

$(x^{2} - 3) (- 8 x) + (x^{2} - 3) (x^{2} + 15) = 0$

Step 2.1.7.2

Factor $x^{2} - 3$ out of $(x^{2} - 3) (- 8 x) + (x^{2} - 3) (x^{2} + 15)$ .

$(x^{2} - 3) (- 8 x + x^{2} + 15) = 0$

Step 2.1.8

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

$(x^{2} - 3) (- 8 u + u^{2} + 15) = 0$

Step 2.1.9

Factor $- 8 u + u^{2} + 15$ using the AC method.

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Step 2.1.9.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 2.1.9.2

Write the factored form using these integers.

$(x^{2} - 3) ((u - 5) (u - 3)) = 0$

Step 2.1.10

Factor.

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Step 2.1.10.1

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

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

Step 2.1.10.2

Remove unnecessary parentheses.

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

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 $0$ .

$x^{2} - 3 = 0$

$x - 5 = 0$

$x - 3 = 0$

Step 2.3

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

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Step 2.3.1

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

$x^{2} - 3 = 0$

Step 2.3.2

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

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Step 2.3.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 2.3.2.2

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

$x = \pm \sqrt{3}$

Step 2.3.2.3

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

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Step 2.3.2.3.1

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

$x = \sqrt{3}$

Step 2.3.2.3.2

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

$x = - \sqrt{3}$

Step 2.3.2.3.3

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

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

Step 2.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.5

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

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Step 2.5.1

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

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

The final solution is all the values that make $(x^{2} - 3) (x - 5) (x - 3) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = \sqrt{3}$ (Multiplicity of $1$ )

$x = - \sqrt{3}$ (Multiplicity of $1$ )

$x = 5$ (Multiplicity of $1$ )

$x = 3$ (Multiplicity of $1$ )

$x = \sqrt{3}$ (Multiplicity of $1$ )

$x = - \sqrt{3}$ (Multiplicity of $1$ )

$x = 5$ (Multiplicity of $1$ )

$x = 3$ (Multiplicity of $1$ )

Step 3