Algebra Examples

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

Step 1

Rewrite the equation as $3 x^{2} - x - 2 = 2$ .

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

Step 2

Subtract $2$ from both sides of the equation.

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

Step 3

Subtract $2$ from $- 2$ .

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

Step 4

Factor by grouping.

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Step 4.1

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 $b = - 1$ .

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Step 4.1.1

Factor $- 1$ out of $- x$ .

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

Step 4.1.2

Rewrite $- 1$ as $3$ plus $- 4$

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2

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

$3 x (x + 1) - 4 (x + 1) = 0$

Step 4.3

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

$(x + 1) (3 x - 4) = 0$

Step 5

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

$x + 1 = 0$

$3 x - 4 = 0$

Step 6

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

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Step 6.1

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

$x + 1 = 0$

Step 6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 7.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 7.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 7.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 7.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 8

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

$x = - 1, \frac{4}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 1, \frac{4}{3}$

Decimal Form:

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

Mixed Number Form:

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