Finite Math Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

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

Step 1.3

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

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

Step 2

Subtract $\frac{x^{2}}{3}$ from both sides of the equation.

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

Step 3

Multiply each term in $\frac{1}{3} + \frac{x}{2} - \frac{x^{2}}{3} = 0$ by $6$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{1}{3} + \frac{x}{2} - \frac{x^{2}}{3} = 0$ by $6$ .

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1.1

Factor $3$ out of $6$ .

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

Step 3.2.1.1.2

Cancel the common factor.

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

Step 3.2.1.1.3

Rewrite the expression.

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

Step 3.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.2.1

Factor $2$ out of $6$ .

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

Step 3.2.1.2.2

Cancel the common factor.

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

Step 3.2.1.2.3

Rewrite the expression.

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

Step 3.2.1.3

Move $3$ to the left of $x$ .

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

Step 3.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

Factor $3$ out of $6$ .

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

Step 3.2.1.4.3

Cancel the common factor.

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

Step 3.2.1.4.4

Rewrite the expression.

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

Step 3.2.1.5

Multiply $2$ by $- 1$ .

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $0$ by $6$ .

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

Step 4

Factor the left side of the equation.

Tap for more steps...

Step 4.1

Factor $- 1$ out of $2 + 3 x - 2 x^{2}$ .

Tap for more steps...

Step 4.1.1

Reorder the expression.

Tap for more steps...

Step 4.1.1.1

Move $2$ .

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

Step 4.1.1.2

Reorder $3 x$ and $- 2 x^{2}$ .

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

Step 4.1.2

Factor $- 1$ out of $- 2 x^{2}$ .

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

Step 4.1.3

Factor $- 1$ out of $3 x$ .

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

Step 4.1.4

Rewrite $2$ as $- 1 (- 2)$ .

$- (2 x^{2}) - (- 3 x) - 1 \cdot - 2 = 0$

Step 4.1.5

Factor $- 1$ out of $- (2 x^{2}) - (- 3 x)$ .

$- (2 x^{2} - 3 x) - 1 \cdot - 2 = 0$

Step 4.1.6

Factor $- 1$ out of $- (2 x^{2} - 3 x) - 1 (- 2)$ .

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

Step 4.2

Factor.

Tap for more steps...

Step 4.2.1

Factor by grouping.

Tap for more steps...

Step 4.2.1.1

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

Tap for more steps...

Step 4.2.1.1.1

Factor $- 3$ out of $- 3 x$ .

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

Step 4.2.1.1.2

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

$- (2 x^{2} + (1 - 4) x - 2) = 0$

Step 4.2.1.1.3

Apply the distributive property.

$- (2 x^{2} + 1 x - 4 x - 2) = 0$

Step 4.2.1.1.4

Multiply $x$ by $1$ .

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

Step 4.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.2.1.2.1

Group the first two terms and the last two terms.

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

Step 4.2.1.2.2

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

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

Step 4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 4.2.2

Remove unnecessary parentheses.

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

Step 5

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

$2 x + 1 = 0$

$x - 2 = 0$

Step 6

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

Tap for more steps...

Step 6.1

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

$2 x + 1 = 0$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 6.2.2

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

Tap for more steps...

Step 6.2.2.1

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

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

Step 6.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.2.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.2.3.1

Move the negative in front of the fraction.

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

Step 7

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

Tap for more steps...

Step 7.1

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

$x - 2 = 0$

Step 7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 8

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

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.5, 2$