Calculus Examples

$3 x^{2} - 4 x - 4 = 0$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = - 4$ .

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Step 1.1.1

Factor $- 4$ out of $- 4 x$ .

$3 x^{2} - 4 x - 4 = 0$

Step 1.1.2

Rewrite $- 4$ as $2$ plus $- 6$

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2

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

$x (3 x + 2) - 2 (3 x + 2) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

$(3 x + 2) (x - 2) = 0$

Step 2

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

$3 x + 2 = 0$

$x - 2 = 0$

Step 3

Set $3 x + 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Step 3.2

Solve $3 x + 2 = 0$ for $x$ .

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Step 3.2.1

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Step 3.2.2

Divide each term in $3 x = - 2$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

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.

$\frac{3 x}{3} = \frac{- 2}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

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.

$x = - \frac{2}{3}$

Step 4

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5

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

$x = - \frac{2}{3}, 2$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{3}, 2$

Decimal Form:

$x = - 0.\overline{6}, 2$