Pre-Algebra Examples

$\frac{1}{3} x^{2} - \frac{1}{18} x - \frac{11}{9} = 0$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{x^{2}}{3} - \frac{1}{18} x - \frac{11}{9} = 0$

Step 1.2

Combine $x$ and $\frac{1}{18}$ .

$\frac{x^{2}}{3} - \frac{x}{18} - \frac{11}{9} = 0$

Step 2

Multiply each term in $\frac{x^{2}}{3} - \frac{x}{18} - \frac{11}{9} = 0$ by $18$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x^{2}}{3} - \frac{x}{18} - \frac{11}{9} = 0$ by $18$ .

$\frac{x^{2}}{3} \cdot 18 - \frac{x}{18} \cdot 18 - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Factor $3$ out of $18$ .

$\frac{x^{2}}{3} \cdot (3 (6)) - \frac{x}{18} \cdot 18 - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{x^{2}}{3} \cdot (3 \cdot 6) - \frac{x}{18} \cdot 18 - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.1.3

Rewrite the expression.

$x^{2} \cdot 6 - \frac{x}{18} \cdot 18 - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.2

Move $6$ to the left of $x^{2}$ .

$6 \cdot x^{2} - \frac{x}{18} \cdot 18 - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.3

Cancel the common factor of $18$ .

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Step 2.2.1.3.1

Move the leading negative in $- \frac{x}{18}$ into the numerator.

$6 x^{2} + \frac{- x}{18} \cdot 18 - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.3.2

Cancel the common factor.

$6 x^{2} + \frac{- x}{18} \cdot 18 - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.3.3

Rewrite the expression.

$6 x^{2} - x - \frac{11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.4

Cancel the common factor of $9$ .

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Step 2.2.1.4.1

Move the leading negative in $- \frac{11}{9}$ into the numerator.

$6 x^{2} - x + \frac{- 11}{9} \cdot 18 = 0 \cdot 18$

Step 2.2.1.4.2

Factor $9$ out of $18$ .

$6 x^{2} - x + \frac{- 11}{9} \cdot (9 (2)) = 0 \cdot 18$

Step 2.2.1.4.3

Cancel the common factor.

$6 x^{2} - x + \frac{- 11}{9} \cdot (9 \cdot 2) = 0 \cdot 18$

Step 2.2.1.4.4

Rewrite the expression.

$6 x^{2} - x - 11 \cdot 2 = 0 \cdot 18$

Step 2.2.1.5

Multiply $- 11$ by $2$ .

$6 x^{2} - x - 22 = 0 \cdot 18$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $18$ .

$6 x^{2} - x - 22 = 0$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 6 \cdot - 22 = - 132$ and whose sum is $b = - 1$ .

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Step 3.1.1

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

$6 x^{2} - (x) - 22 = 0$

Step 3.1.2

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

$6 x^{2} + (11 - 12) x - 22 = 0$

Step 3.1.3

Apply the distributive property.

$6 x^{2} + 11 x - 12 x - 22 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(6 x^{2} + 11 x) - 12 x - 22 = 0$

Step 3.2.2

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

$x (6 x + 11) - 2 (6 x + 11) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $6 x + 11$ .

$(6 x + 11) (x - 2) = 0$

Step 4

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

$6 x + 11 = 0$

$x - 2 = 0$

Step 5

Set $6 x + 11$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $6 x + 11$ equal to $0$ .

$6 x + 11 = 0$

Step 5.2

Solve $6 x + 11 = 0$ for $x$ .

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Step 5.2.1

Subtract $11$ from both sides of the equation.

$6 x = - 11$

Step 5.2.2

Divide each term in $6 x = - 11$ by $6$ and simplify.

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Step 5.2.2.1

Divide each term in $6 x = - 11$ by $6$ .

$\frac{6 x}{6} = \frac{- 11}{6}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 11}{6}$

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{6}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{6}$

Step 6

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

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Step 6.1

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

$x - 2 = 0$

Step 6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7

The final solution is all the values that make $(6 x + 11) (x - 2) = 0$ true.

$x = - \frac{11}{6}, 2$