Calculus Examples

$12 x^{2} - 66 x - 36 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $6$ out of $12 x^{2} - 66 x - 36$ .

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Step 1.1.1

Factor $6$ out of $12 x^{2}$ .

$6 (2 x^{2}) - 66 x - 36 = 0$

Step 1.1.2

Factor $6$ out of $- 66 x$ .

$6 (2 x^{2}) + 6 (- 11 x) - 36 = 0$

Step 1.1.3

Factor $6$ out of $- 36$ .

$6 (2 x^{2}) + 6 (- 11 x) + 6 (- 6) = 0$

Step 1.1.4

Factor $6$ out of $6 (2 x^{2}) + 6 (- 11 x)$ .

$6 (2 x^{2} - 11 x) + 6 (- 6) = 0$

Step 1.1.5

Factor $6$ out of $6 (2 x^{2} - 11 x) + 6 (- 6)$ .

$6 (2 x^{2} - 11 x - 6) = 0$

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 11$ .

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Step 1.2.1.1.1

Factor $- 11$ out of $- 11 x$ .

$6 (2 x^{2} - 11 x - 6) = 0$

Step 1.2.1.1.2

Rewrite $- 11$ as $1$ plus $- 12$

$6 (2 x^{2} + (1 - 12) x - 6) = 0$

Step 1.2.1.1.3

Apply the distributive property.

$6 (2 x^{2} + 1 x - 12 x - 6) = 0$

Step 1.2.1.1.4

Multiply $x$ by $1$ .

$6 (2 x^{2} + x - 12 x - 6) = 0$

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.

$6 ((2 x^{2} + x) - 12 x - 6) = 0$

Step 1.2.1.2.2

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

$6 (x (2 x + 1) - 6 (2 x + 1)) = 0$

Step 1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$6 ((2 x + 1) (x - 6)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$6 (2 x + 1) (x - 6) = 0$

Step 2

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

$2 x + 1 = 0$

$x - 6 = 0$

Step 3

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 3.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 3.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 3.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

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.

$x = - \frac{1}{2}$

Step 4

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 5

The final solution is all the values that make $6 (2 x + 1) (x - 6) = 0$ true.

$x = - \frac{1}{2}, 6$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, 6$

Decimal Form:

$x = - 0.5, 6$