Pre-Algebra Examples

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

Step 1

Simplify $\frac{1}{3} x^{2}$ .

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

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

Step 1.3

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

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

Step 2

Combine $x$ and $\frac{1}{6}$ .

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

Step 3

Add $\frac{x}{6}$ to both sides of the equation.

$\frac{x^{2}}{3} + \frac{x}{6} = \frac{1}{2}$

Step 4

Multiply each term in $\frac{x^{2}}{3} + \frac{x}{6} = \frac{1}{2}$ by $6$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{x^{2}}{3} + \frac{x}{6} = \frac{1}{2}$ by $6$ .

$\frac{x^{2}}{3} \cdot 6 + \frac{x}{6} \cdot 6 = \frac{1}{2} \cdot 6$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $6$ .

$\frac{x^{2}}{3} \cdot (3 (2)) + \frac{x}{6} \cdot 6 = \frac{1}{2} \cdot 6$

Step 4.2.1.1.2

Cancel the common factor.

$\frac{x^{2}}{3} \cdot (3 \cdot 2) + \frac{x}{6} \cdot 6 = \frac{1}{2} \cdot 6$

Step 4.2.1.1.3

Rewrite the expression.

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Cancel the common factor of $6$ .

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Step 4.2.1.3.1

Cancel the common factor.

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

Step 4.2.1.3.2

Rewrite the expression.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Factor $2$ out of $6$ .

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

Step 4.3.1.2

Cancel the common factor.

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

Step 4.3.1.3

Rewrite the expression.

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

Step 5

Subtract $3$ from both sides of the equation.

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

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

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Step 6.1.1

Multiply by $1$ .

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

Step 6.1.2

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

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

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

Step 6.2.2

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

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

Step 6.3

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

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

Step 7

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

$x - 1 = 0$

$2 x + 3 = 0$

Step 8

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

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Step 8.1

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

$x - 1 = 0$

Step 8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9

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

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Step 9.1

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

$2 x + 3 = 0$

Step 9.2

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

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Step 9.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 9.2.2

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

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Step 9.2.2.1

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

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

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $2$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

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

Step 9.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 9.2.2.3

Simplify the right side.

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Step 9.2.2.3.1

Move the negative in front of the fraction.

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

Step 10

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

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