Pre-Algebra Examples

$\frac{1}{6} x + \frac{2}{3} = \frac{1}{4} \cdot (x (x - 2))$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{1}{4} \cdot (x (x - 2)) = \frac{1}{6} x + \frac{2}{3}$

Step 2

Simplify $\frac{1}{4} \cdot (x (x - 2))$ .

Tap for more steps...

Step 2.1

Rewrite.

$0 + 0 + \frac{1}{4} \cdot (x (x - 2)) = \frac{1}{6} x + \frac{2}{3}$

Step 2.2

Simplify by adding zeros.

$\frac{1}{4} \cdot (x (x - 2)) = \frac{1}{6} x + \frac{2}{3}$

Step 2.3

Apply the distributive property.

$\frac{1}{4} \cdot (x \cdot x + x \cdot - 2) = \frac{1}{6} x + \frac{2}{3}$

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

Multiply $x$ by $x$ .

$\frac{1}{4} \cdot (x^{2} + x \cdot - 2) = \frac{1}{6} x + \frac{2}{3}$

Step 2.4.2

Move $- 2$ to the left of $x$ .

$\frac{1}{4} \cdot (x^{2} - 2 \cdot x) = \frac{1}{6} x + \frac{2}{3}$

Step 2.5

Apply the distributive property.

$\frac{1}{4} x^{2} + \frac{1}{4} (- 2 x) = \frac{1}{6} x + \frac{2}{3}$

Step 2.6

Combine $\frac{1}{4}$ and $x^{2}$ .

$\frac{x^{2}}{4} + \frac{1}{4} (- 2 x) = \frac{1}{6} x + \frac{2}{3}$

Step 2.7

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.7.1

Factor $2$ out of $4$ .

$\frac{x^{2}}{4} + \frac{1}{2 (2)} (- 2 x) = \frac{1}{6} x + \frac{2}{3}$

Step 2.7.2

Factor $2$ out of $- 2 x$ .

$\frac{x^{2}}{4} + \frac{1}{2 (2)} (2 (- x)) = \frac{1}{6} x + \frac{2}{3}$

Step 2.7.3

Cancel the common factor.

$\frac{x^{2}}{4} + \frac{1}{2 \cdot 2} (2 (- x)) = \frac{1}{6} x + \frac{2}{3}$

Step 2.7.4

Rewrite the expression.

$\frac{x^{2}}{4} + \frac{1}{2} (- x) = \frac{1}{6} x + \frac{2}{3}$

Step 2.8

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

$\frac{x^{2}}{4} - \frac{x}{2} = \frac{1}{6} x + \frac{2}{3}$

Step 3

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

$\frac{x^{2}}{4} - \frac{x}{2} = \frac{x}{6} + \frac{2}{3}$

Step 4

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 4.1

Subtract $\frac{x}{6}$ from both sides of the equation.

$\frac{x^{2}}{4} - \frac{x}{2} - \frac{x}{6} = \frac{2}{3}$

Step 4.2

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x^{2}}{4} - \frac{x}{2} \cdot \frac{3}{3} - \frac{x}{6} = \frac{2}{3}$

Step 4.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.3.1

Multiply $\frac{x}{2}$ by $\frac{3}{3}$ .

$\frac{x^{2}}{4} - \frac{x \cdot 3}{2 \cdot 3} - \frac{x}{6} = \frac{2}{3}$

Step 4.3.2

Multiply $2$ by $3$ .

$\frac{x^{2}}{4} - \frac{x \cdot 3}{6} - \frac{x}{6} = \frac{2}{3}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{x^{2}}{4} + \frac{- x \cdot 3 - x}{6} = \frac{2}{3}$

Step 4.5

Simplify each term.

Tap for more steps...

Step 4.5.1

Simplify the numerator.

Tap for more steps...

Step 4.5.1.1

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

Tap for more steps...

Step 4.5.1.1.1

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

$\frac{x^{2}}{4} + \frac{x (- 1 \cdot 3) - x}{6} = \frac{2}{3}$

Step 4.5.1.1.2

Factor $x$ out of $- x$ .

$\frac{x^{2}}{4} + \frac{x (- 1 \cdot 3) + x \cdot - 1}{6} = \frac{2}{3}$

Step 4.5.1.1.3

Factor $x$ out of $x (- 1 \cdot 3) + x \cdot - 1$ .

$\frac{x^{2}}{4} + \frac{x (- 1 \cdot 3 - 1)}{6} = \frac{2}{3}$

Step 4.5.1.2

Multiply $- 1$ by $3$ .

$\frac{x^{2}}{4} + \frac{x (- 3 - 1)}{6} = \frac{2}{3}$

Step 4.5.1.3

Subtract $1$ from $- 3$ .

$\frac{x^{2}}{4} + \frac{x \cdot - 4}{6} = \frac{2}{3}$

Step 4.5.2

Cancel the common factor of $- 4$ and $6$ .

Tap for more steps...

Step 4.5.2.1

Factor $2$ out of $x \cdot - 4$ .

$\frac{x^{2}}{4} + \frac{2 (x \cdot - 2)}{6} = \frac{2}{3}$

Step 4.5.2.2

Cancel the common factors.

Tap for more steps...

Step 4.5.2.2.1

Factor $2$ out of $6$ .

$\frac{x^{2}}{4} + \frac{2 (x \cdot - 2)}{2 (3)} = \frac{2}{3}$

Step 4.5.2.2.2

Cancel the common factor.

$\frac{x^{2}}{4} + \frac{2 (x \cdot - 2)}{2 \cdot 3} = \frac{2}{3}$

Step 4.5.2.2.3

Rewrite the expression.

$\frac{x^{2}}{4} + \frac{x \cdot - 2}{3} = \frac{2}{3}$

Step 4.5.3

Move $- 2$ to the left of $x$ .

$\frac{x^{2}}{4} + \frac{- 2 \cdot x}{3} = \frac{2}{3}$

Step 4.5.4

Move the negative in front of the fraction.

$\frac{x^{2}}{4} - \frac{2 x}{3} = \frac{2}{3}$

Step 5

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

$\frac{x^{2}}{4} - \frac{2 x}{3} - \frac{2}{3} = 0$

Step 6

Multiply through by the least common denominator $12$ , then simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$12 (\frac{x^{2}}{4}) + 12 (- \frac{2 x}{3}) + 12 (- \frac{2}{3}) = 0$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.1.1

Factor $4$ out of $12$ .

$4 (3) (\frac{x^{2}}{4}) + 12 (- \frac{2 x}{3}) + 12 (- \frac{2}{3}) = 0$

Step 6.2.1.2

Cancel the common factor.

$4 \cdot (3 (\frac{x^{2}}{4})) + 12 (- \frac{2 x}{3}) + 12 (- \frac{2}{3}) = 0$

Step 6.2.1.3

Rewrite the expression.

$3 x^{2} + 12 (- \frac{2 x}{3}) + 12 (- \frac{2}{3}) = 0$

Step 6.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.2.1

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

$3 x^{2} + 12 (\frac{- 2 x}{3}) + 12 (- \frac{2}{3}) = 0$

Step 6.2.2.2

Factor $3$ out of $12$ .

$3 x^{2} + 3 (4) (\frac{- 2 x}{3}) + 12 (- \frac{2}{3}) = 0$

Step 6.2.2.3

Cancel the common factor.

$3 x^{2} + 3 \cdot (4 (\frac{- 2 x}{3})) + 12 (- \frac{2}{3}) = 0$

Step 6.2.2.4

Rewrite the expression.

$3 x^{2} + 4 (- 2 x) + 12 (- \frac{2}{3}) = 0$

Step 6.2.3

Multiply $- 2$ by $4$ .

$3 x^{2} - 8 x + 12 (- \frac{2}{3}) = 0$

Step 6.2.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.4.1

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

$3 x^{2} - 8 x + 12 (\frac{- 2}{3}) = 0$

Step 6.2.4.2

Factor $3$ out of $12$ .

$3 x^{2} - 8 x + 3 (4) (\frac{- 2}{3}) = 0$

Step 6.2.4.3

Cancel the common factor.

$3 x^{2} - 8 x + 3 \cdot (4 (\frac{- 2}{3})) = 0$

Step 6.2.4.4

Rewrite the expression.

$3 x^{2} - 8 x + 4 \cdot - 2 = 0$

Step 6.2.5

Multiply $4$ by $- 2$ .

$3 x^{2} - 8 x - 8 = 0$

Step 7

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8

Substitute the values $a = 3$ , $b = - 8$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot - 8)}}{2 \cdot 3}$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 8}}{2 \cdot 3}$

Step 9.1.2

Multiply $- 4 \cdot 3 \cdot - 8$ .

Tap for more steps...

Step 9.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot - 8}}{2 \cdot 3}$

Step 9.1.2.2

Multiply $- 12$ by $- 8$ .

$x = \frac{8 \pm \sqrt{64 + 96}}{2 \cdot 3}$

Step 9.1.3

Add $64$ and $96$ .

$x = \frac{8 \pm \sqrt{160}}{2 \cdot 3}$

Step 9.1.4

Rewrite $160$ as $4^{2} \cdot 10$ .

Tap for more steps...

Step 9.1.4.1

Factor $16$ out of $160$ .

$x = \frac{8 \pm \sqrt{16 (10)}}{2 \cdot 3}$

Step 9.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 3}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{10}}{2 \cdot 3}$

Step 9.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 4 \sqrt{10}}{6}$

Step 9.3

Simplify $\frac{8 \pm 4 \sqrt{10}}{6}$ .

$x = \frac{4 \pm 2 \sqrt{10}}{3}$

Step 10

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 8}}{2 \cdot 3}$

Step 10.1.2

Multiply $- 4 \cdot 3 \cdot - 8$ .

Tap for more steps...

Step 10.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot - 8}}{2 \cdot 3}$

Step 10.1.2.2

Multiply $- 12$ by $- 8$ .

$x = \frac{8 \pm \sqrt{64 + 96}}{2 \cdot 3}$

Step 10.1.3

Add $64$ and $96$ .

$x = \frac{8 \pm \sqrt{160}}{2 \cdot 3}$

Step 10.1.4

Rewrite $160$ as $4^{2} \cdot 10$ .

Tap for more steps...

Step 10.1.4.1

Factor $16$ out of $160$ .

$x = \frac{8 \pm \sqrt{16 (10)}}{2 \cdot 3}$

Step 10.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 3}$

Step 10.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{10}}{2 \cdot 3}$

Step 10.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 4 \sqrt{10}}{6}$

Step 10.3

Simplify $\frac{8 \pm 4 \sqrt{10}}{6}$ .

$x = \frac{4 \pm 2 \sqrt{10}}{3}$

Step 10.4

Change the $\pm$ to $+$ .

$x = \frac{4 + 2 \sqrt{10}}{3}$

Step 11

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 11.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot - 8}}{2 \cdot 3}$

Step 11.1.2

Multiply $- 4 \cdot 3 \cdot - 8$ .

Tap for more steps...

Step 11.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot - 8}}{2 \cdot 3}$

Step 11.1.2.2

Multiply $- 12$ by $- 8$ .

$x = \frac{8 \pm \sqrt{64 + 96}}{2 \cdot 3}$

Step 11.1.3

Add $64$ and $96$ .

$x = \frac{8 \pm \sqrt{160}}{2 \cdot 3}$

Step 11.1.4

Rewrite $160$ as $4^{2} \cdot 10$ .

Tap for more steps...

Step 11.1.4.1

Factor $16$ out of $160$ .

$x = \frac{8 \pm \sqrt{16 (10)}}{2 \cdot 3}$

Step 11.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 3}$

Step 11.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{10}}{2 \cdot 3}$

Step 11.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 4 \sqrt{10}}{6}$

Step 11.3

Simplify $\frac{8 \pm 4 \sqrt{10}}{6}$ .

$x = \frac{4 \pm 2 \sqrt{10}}{3}$

Step 11.4

Change the $\pm$ to $-$ .

$x = \frac{4 - 2 \sqrt{10}}{3}$

Step 12

The final answer is the combination of both solutions.

$x = \frac{4 + 2 \sqrt{10}}{3}, \frac{4 - 2 \sqrt{10}}{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 + 2 \sqrt{10}}{3}, \frac{4 - 2 \sqrt{10}}{3}$

Decimal Form:

$x = 3.44151844 \dots, - 0.77485177 \dots$