Algebra Examples

$f (x) = \frac{6 x^{2} - x - 1}{x + 2}$

Step 1

Set $\frac{6 x^{2} - x - 1}{x + 2}$ equal to $0$ .

$\frac{6 x^{2} - x - 1}{x + 2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Set the numerator equal to zero.

$6 x^{2} - x - 1 = 0$

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

Factor by grouping.

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Step 2.2.1.1

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 $a \cdot c = 6 \cdot - 1 = - 6$ and whose sum is $b = - 1$ .

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

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

$6 x^{2} + (2 - 3) x - 1 = 0$

Step 2.2.1.1.3

Apply the distributive property.

$6 x^{2} + 2 x - 3 x - 1 = 0$

Step 2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.1.2.1

Group the first two terms and the last two terms.

$(6 x^{2} + 2 x) - 3 x - 1 = 0$

Step 2.2.1.2.2

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

$2 x (3 x + 1) - (3 x + 1) = 0$

Step 2.2.1.3

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

$(3 x + 1) (2 x - 1) = 0$

Step 2.2.2

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

$3 x + 1 = 0$

$2 x - 1 = 0$

Step 2.2.3

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

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Step 2.2.3.1

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

$3 x + 1 = 0$

Step 2.2.3.2

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

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Step 2.2.3.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 2.2.3.2.2

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

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Step 2.2.3.2.2.1

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

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

Step 2.2.3.2.2.2

Simplify the left side.

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Step 2.2.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.3.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3.2.2.3

Simplify the right side.

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Step 2.2.3.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.4

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

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Step 2.2.4.1

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

$2 x - 1 = 0$

Step 2.2.4.2

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

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Step 2.2.4.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.2.4.2.2

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

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Step 2.2.4.2.2.1

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

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

Step 2.2.4.2.2.2

Simplify the left side.

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Step 2.2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.5

The final solution is all the values that make $(3 x + 1) (2 x - 1) = 0$ true.

$x = - \frac{1}{3}, \frac{1}{2}$

Step 3