Algebra Examples

$f (x) = \frac{1}{3} x^{2} + \frac{1}{3} x - \frac{2}{3}$

Step 1

Set $\frac{1}{3} x^{2} + \frac{1}{3} x - \frac{2}{3}$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{1}{3}$ and $x^{2}$ .

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

Step 2.1.2

Combine $\frac{1}{3}$ and $x$ .

$\frac{x^{2}}{3} + \frac{x}{3} - \frac{2}{3} = 0$

Step 2.2

Multiply each term in $\frac{x^{2}}{3} + \frac{x}{3} - \frac{2}{3} = 0$ by $3$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $\frac{x^{2}}{3} + \frac{x}{3} - \frac{2}{3} = 0$ by $3$ .

$\frac{x^{2}}{3} \cdot 3 + \frac{x}{3} \cdot 3 - \frac{2}{3} \cdot 3 = 0 \cdot 3$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1.1

Cancel the common factor.

$\frac{x^{2}}{3} \cdot 3 + \frac{x}{3} \cdot 3 - \frac{2}{3} \cdot 3 = 0 \cdot 3$

Step 2.2.2.1.1.2

Rewrite the expression.

$x^{2} + \frac{x}{3} \cdot 3 - \frac{2}{3} \cdot 3 = 0 \cdot 3$

Step 2.2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.2.1

Cancel the common factor.

$x^{2} + \frac{x}{3} \cdot 3 - \frac{2}{3} \cdot 3 = 0 \cdot 3$

Step 2.2.2.1.2.2

Rewrite the expression.

$x^{2} + x - \frac{2}{3} \cdot 3 = 0 \cdot 3$

Step 2.2.2.1.3

Cancel the common factor of $3$ .

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Step 2.2.2.1.3.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$x^{2} + x + \frac{- 2}{3} \cdot 3 = 0 \cdot 3$

Step 2.2.2.1.3.2

Cancel the common factor.

$x^{2} + x + \frac{- 2}{3} \cdot 3 = 0 \cdot 3$

Step 2.2.2.1.3.3

Rewrite the expression.

$x^{2} + x - 2 = 0 \cdot 3$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $3$ .

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

Step 2.3

Factor $x^{2} + x - 2$ using the AC method.

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Step 2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 2.3.2

Write the factored form using these integers.

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

Step 2.4

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

$x - 1 = 0$

$x + 2 = 0$

Step 2.5

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

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Step 2.5.1

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

$x - 1 = 0$

Step 2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.6

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

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Step 2.6.1

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

$x + 2 = 0$

Step 2.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.7

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

$x = 1, - 2$

Step 3