Calculus Examples

f (x) = 3 x^{3} - 10 x^{2} + 3 x

$f (x) = 3 x^{3} - 10 x^{2} + 3 x$

Step 1

Set

3 x^{3} - 10 x^{2} + 3 x

$3 x^{3} - 10 x^{2} + 3 x$ equal to

0

$0$ .

3 x^{3} - 10 x^{2} + 3 x = 0

$3 x^{3} - 10 x^{2} + 3 x = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor

x

$x$ out of

3 x^{3} - 10 x^{2} + 3 x

$3 x^{3} - 10 x^{2} + 3 x$ .

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Step 2.1.1.1

Factor

x

$x$ out of

3 x^{3}

$3 x^{3}$ .

x (3 x^{2}) - 10 x^{2} + 3 x = 0

$x (3 x^{2}) - 10 x^{2} + 3 x = 0$

Step 2.1.1.2

Factor

x

$x$ out of

- 10 x^{2}

$- 10 x^{2}$ .

x (3 x^{2}) + x (- 10 x) + 3 x = 0

$x (3 x^{2}) + x (- 10 x) + 3 x = 0$

Step 2.1.1.3

Factor

x

$x$ out of

3 x

$3 x$ .

x (3 x^{2}) + x (- 10 x) + x \cdot 3 = 0

$x (3 x^{2}) + x (- 10 x) + x \cdot 3 = 0$

Step 2.1.1.4

Factor

x

$x$ out of

x (3 x^{2}) + x (- 10 x)

$x (3 x^{2}) + x (- 10 x)$ .

x (3 x^{2} - 10 x) + x \cdot 3 = 0

$x (3 x^{2} - 10 x) + x \cdot 3 = 0$

Step 2.1.1.5

Factor

x

$x$ out of

x (3 x^{2} - 10 x) + x \cdot 3

$x (3 x^{2} - 10 x) + x \cdot 3$ .

x (3 x^{2} - 10 x + 3) = 0

$x (3 x^{2} - 10 x + 3) = 0$

x (3 x^{2} - 10 x + 3) = 0

$x (3 x^{2} - 10 x + 3) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor by grouping.

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Step 2.1.2.1.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 3 \cdot 3 = 9

$a \cdot c = 3 \cdot 3 = 9$ and whose sum is

b = - 10

$b = - 10$ .

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Step 2.1.2.1.1.1

Factor

- 10

$- 10$ out of

- 10 x

$- 10 x$ .

x (3 x^{2} - 10 x + 3) = 0

$x (3 x^{2} - 10 x + 3) = 0$

Step 2.1.2.1.1.2

Rewrite

- 10

$- 10$ as

- 1

$- 1$ plus

- 9

$- 9$

x (3 x^{2} + (- 1 - 9) x + 3) = 0

$x (3 x^{2} + (- 1 - 9) x + 3) = 0$

Step 2.1.2.1.1.3

Apply the distributive property.

x (3 x^{2} - 1 x - 9 x + 3) = 0

$x (3 x^{2} - 1 x - 9 x + 3) = 0$

x (3 x^{2} - 1 x - 9 x + 3) = 0

$x (3 x^{2} - 1 x - 9 x + 3) = 0$

Step 2.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1.2.1

Group the first two terms and the last two terms.

x ((3 x^{2} - 1 x) - 9 x + 3) = 0

$x ((3 x^{2} - 1 x) - 9 x + 3) = 0$

Step 2.1.2.1.2.2

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

x (x (3 x - 1) - 3 (3 x - 1)) = 0

$x (x (3 x - 1) - 3 (3 x - 1)) = 0$

x (x (3 x - 1) - 3 (3 x - 1)) = 0

$x (x (3 x - 1) - 3 (3 x - 1)) = 0$

Step 2.1.2.1.3

Factor the polynomial by factoring out the greatest common factor,

3 x - 1

$3 x - 1$ .

x ((3 x - 1) (x - 3)) = 0

$x ((3 x - 1) (x - 3)) = 0$

x ((3 x - 1) (x - 3)) = 0

$x ((3 x - 1) (x - 3)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

x (3 x - 1) (x - 3) = 0

$x (3 x - 1) (x - 3) = 0$

x (3 x - 1) (x - 3) = 0

$x (3 x - 1) (x - 3) = 0$

x (3 x - 1) (x - 3) = 0

$x (3 x - 1) (x - 3) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

x = 0

$x = 0$

3 x - 1 = 0

$3 x - 1 = 0$

x - 3 = 0

$x - 3 = 0$

Step 2.3

Set

x

$x$ equal to

0

$0$ .

x = 0

$x = 0$

Step 2.4

Set

3 x - 1

$3 x - 1$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.4.1

Set

3 x - 1

$3 x - 1$ equal to

0

$0$ .

3 x - 1 = 0

$3 x - 1 = 0$

Step 2.4.2

Solve

3 x - 1 = 0

$3 x - 1 = 0$ for

x

$x$ .

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Step 2.4.2.1

Add

1

$1$ to both sides of the equation.

3 x = 1

$3 x = 1$

Step 2.4.2.2

Divide each term in

3 x = 1

$3 x = 1$ by

3

$3$ and simplify.

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Step 2.4.2.2.1

Divide each term in

3 x = 1

$3 x = 1$ by

3

$3$ .

\frac{3 x}{3} = \frac{1}{3}

$\frac{3 x}{3} = \frac{1}{3}$

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

3

$3$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

\frac{3 x}{3} = \frac{1}{3}

$\frac{3 x}{3} = \frac{1}{3}$

Step 2.4.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{1}{3}

$x = \frac{1}{3}$

x = \frac{1}{3}

$x = \frac{1}{3}$

x = \frac{1}{3}

$x = \frac{1}{3}$

x = \frac{1}{3}

$x = \frac{1}{3}$

x = \frac{1}{3}

$x = \frac{1}{3}$

x = \frac{1}{3}

$x = \frac{1}{3}$

Step 2.5

Set

x - 3

$x - 3$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.5.1

Set

x - 3

$x - 3$ equal to

0

$0$ .

x - 3 = 0

$x - 3 = 0$

Step 2.5.2

Add

3

$3$ to both sides of the equation.

x = 3

$x = 3$

x = 3

$x = 3$

Step 2.6

The final solution is all the values that make

x (3 x - 1) (x - 3) = 0

$x (3 x - 1) (x - 3) = 0$ true.

x = 0, \frac{1}{3}, 3

$x = 0, \frac{1}{3}, 3$

x = 0, \frac{1}{3}, 3

$x = 0, \frac{1}{3}, 3$

Step 3

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