Finite Math Examples

$y = 3 x^{4} - 8 x^{3} - 6 x^{2} + 24 x - 9$

Step 1

Set $3 x^{4} - 8 x^{3} - 6 x^{2} + 24 x - 9$ equal to $0$ .

$3 x^{4} - 8 x^{3} - 6 x^{2} + 24 x - 9 = 0$

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.

$- 8 x^{3} + 24 x + 3 x^{4} - 6 x^{2} - 9 = 0$

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 $- 8 x^{3}$ .

$- 8 x \cdot x^{2} + 24 x + 3 x^{4} - 6 x^{2} - 9 = 0$

Step 2.1.2.2

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

$- 8 x \cdot x^{2} - 8 x \cdot - 3 + 3 x^{4} - 6 x^{2} - 9 = 0$

Step 2.1.2.3

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

$- 8 x (x^{2} - 3) + 3 x^{4} - 6 x^{2} - 9 = 0$

Step 2.1.3

Factor $3$ out of $3 x^{4} - 6 x^{2} - 9$ .

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Step 2.1.3.1

Factor $3$ out of $3 x^{4}$ .

$- 8 x (x^{2} - 3) + 3 (x^{4}) - 6 x^{2} - 9 = 0$

Step 2.1.3.2

Factor $3$ out of $- 6 x^{2}$ .

$- 8 x (x^{2} - 3) + 3 (x^{4}) + 3 (- 2 x^{2}) - 9 = 0$

Step 2.1.3.3

Factor $3$ out of $- 9$ .

$- 8 x (x^{2} - 3) + 3 x^{4} + 3 (- 2 x^{2}) + 3 \cdot - 3 = 0$

Step 2.1.3.4

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

$- 8 x (x^{2} - 3) + 3 (x^{4} - 2 x^{2}) + 3 \cdot - 3 = 0$

Step 2.1.3.5

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

Factor $u^{2} - 2 u - 3$ using the AC method.

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Step 2.1.6.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 2.1.6.2

Write the factored form using these integers.

$- 8 x (x^{2} - 3) + 3 ((u - 3) (u + 1)) = 0$

Step 2.1.7

Factor.

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Step 2.1.7.1

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

$- 8 x (x^{2} - 3) + 3 ((x^{2} - 3) (x^{2} + 1)) = 0$

Step 2.1.7.2

Remove unnecessary parentheses.

$- 8 x (x^{2} - 3) + 3 (x^{2} - 3) (x^{2} + 1) = 0$

Step 2.1.8

Factor $x^{2} - 3$ out of $- 8 x (x^{2} - 3) + 3 (x^{2} - 3) (x^{2} + 1)$ .

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Step 2.1.8.1

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

$(x^{2} - 3) (- 8 x) + 3 (x^{2} - 3) (x^{2} + 1) = 0$

Step 2.1.8.2

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

$(x^{2} - 3) (- 8 x) + (x^{2} - 3) (3 (x^{2} + 1)) = 0$

Step 2.1.8.3

Factor $x^{2} - 3$ out of $(x^{2} - 3) (- 8 x) + (x^{2} - 3) (3 (x^{2} + 1))$ .

$(x^{2} - 3) (- 8 x + 3 (x^{2} + 1)) = 0$

Step 2.1.9

Apply the distributive property.

$(x^{2} - 3) (- 8 x + 3 x^{2} + 3 \cdot 1) = 0$

Step 2.1.10

Multiply $3$ by $1$ .

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

Step 2.1.11

Reorder terms.

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

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 $0$ .

$x^{2} - 3 = 0$

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

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 $3 x^{2} - 8 x + 3$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $3 x^{2} - 8 x + 3$ equal to $0$ .

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

Step 2.4.2

Solve $3 x^{2} - 8 x + 3 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 3$ , $b = - 8$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot 3)}}{2 \cdot 3}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 3}}{2 \cdot 3}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 3 \cdot 3$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 3}}{2 \cdot 3}$

Step 2.4.2.3.1.2.2

Multiply $- 12$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 36}}{2 \cdot 3}$

Step 2.4.2.3.1.3

Subtract $36$ from $64$ .

$x = \frac{8 \pm \sqrt{28}}{2 \cdot 3}$

Step 2.4.2.3.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.4.2.3.1.4.1

Factor $4$ out of $28$ .

$x = \frac{8 \pm \sqrt{4 (7)}}{2 \cdot 3}$

Step 2.4.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 3}$

Step 2.4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{7}}{2 \cdot 3}$

Step 2.4.2.3.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{7}}{6}$

Step 2.4.2.3.3

Simplify $\frac{8 \pm 2 \sqrt{7}}{6}$ .

$x = \frac{4 \pm \sqrt{7}}{3}$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{7}}{3}, \frac{4 - \sqrt{7}}{3}$

Step 2.5

The final solution is all the values that make $(x^{2} - 3) (3 x^{2} - 8 x + 3) = 0$ true.

$x = \sqrt{3}, - \sqrt{3}, \frac{4 + \sqrt{7}}{3}, \frac{4 - \sqrt{7}}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3}, - \sqrt{3}, \frac{4 + \sqrt{7}}{3}, \frac{4 - \sqrt{7}}{3}$

Decimal Form:

$x = 1.73205080 \dots, - 1.73205080 \dots, 2.21525043 \dots, 0.45141622 \dots$

Step 4