Finite Math Examples

$f (x) = 3 x^{3} - 12 x^{2} + 3 x$

Step 1

Set $3 x^{3} - 12 x^{2} + 3 x$ equal to $0$ .

$3 x^{3} - 12 x^{2} + 3 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $3 x$ out of $3 x^{3} - 12 x^{2} + 3 x$ .

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Step 2.1.1

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

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

Step 2.1.2

Factor $3 x$ out of $- 12 x^{2}$ .

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

Step 2.1.3

Factor $3 x$ out of $3 x$ .

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

Step 2.1.4

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

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

Step 2.1.5

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

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

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 $0$ .

$x = 0$

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

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Solve $x^{2} - 4 x + 1 = 0$ 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 $a = 1$ , $b = - 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

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 $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.3.1.2

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

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Subtract $4$ from $16$ .

$x = \frac{4 \pm \sqrt{12}}{2 \cdot 1}$

Step 2.4.2.3.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.4.2.3.1.4.1

Factor $4$ out of $12$ .

$x = \frac{4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.4.2.3.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{3}}{2}$

Step 2.4.2.3.3

Simplify $\frac{4 \pm 2 \sqrt{3}}{2}$ .

$x = 2 \pm \sqrt{3}$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = 2 + \sqrt{3}, 2 - \sqrt{3}$

Step 2.5

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

$x = 0, 2 + \sqrt{3}, 2 - \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, 2 + \sqrt{3}, 2 - \sqrt{3}$

Decimal Form:

$x = 0, 3.73205080 \dots, 0.26794919 \dots$

Step 4