Calculus Examples

$9 x^{2} - 6 x - 3 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

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Step 1.1.1

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

$3 (3 x^{2}) - 6 x - 3 = 0$

Step 1.1.2

Factor $3$ out of $- 6 x$ .

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

Step 1.1.3

Factor $3$ out of $- 3$ .

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.2

Factor.

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Step 1.2.1

Factor by grouping.

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Step 1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Step 1.2.1.1.1

Factor $- 2$ out of $- 2 x$ .

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

Step 1.2.1.1.2

Rewrite $- 2$ as $1$ plus $- 3$

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

Step 1.2.1.1.3

Apply the distributive property.

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

Step 1.2.1.1.4

Multiply $x$ by $1$ .

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

Step 1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.1.2.2

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

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

Step 1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

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

Step 1.2.2

Remove unnecessary parentheses.

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

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x + 1 = 0$

$x - 1 = 0$

Step 3

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 3.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 3.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

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

Step 4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.\overline{3}, 1$