Algebra Examples

$\frac{1}{2} w^{2} - \frac{2}{3} = \frac{5}{6} w$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Combine $\frac{1}{2}$ and $w^{2}$ .

$\frac{w^{2}}{2} - \frac{2}{3} = \frac{5}{6} w$

Step 1.2

Simplify the right side.

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Step 1.2.1

Combine $\frac{5}{6}$ and $w$ .

$\frac{w^{2}}{2} - \frac{2}{3} = \frac{5 w}{6}$

Step 1.3

Subtract $\frac{5 w}{6}$ from both sides of the equation.

$\frac{w^{2}}{2} - \frac{2}{3} - \frac{5 w}{6} = 0$

Step 2

Multiply through by the least common denominator $6$ , then simplify.

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Step 2.1

Apply the distributive property.

$6 (\frac{w^{2}}{2}) + 6 (- \frac{2}{3}) + 6 (- \frac{5 w}{6}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Factor $2$ out of $6$ .

$2 (3) (\frac{w^{2}}{2}) + 6 (- \frac{2}{3}) + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.1.2

Cancel the common factor.

$2 \cdot (3 (\frac{w^{2}}{2})) + 6 (- \frac{2}{3}) + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.1.3

Rewrite the expression.

$3 w^{2} + 6 (- \frac{2}{3}) + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

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

$3 w^{2} + 6 (\frac{- 2}{3}) + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.2.2

Factor $3$ out of $6$ .

$3 w^{2} + 3 (2) (\frac{- 2}{3}) + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.2.3

Cancel the common factor.

$3 w^{2} + 3 \cdot (2 (\frac{- 2}{3})) + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.2.4

Rewrite the expression.

$3 w^{2} + 2 \cdot - 2 + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.3

Multiply $2$ by $- 2$ .

$3 w^{2} - 4 + 6 (- \frac{5 w}{6}) = 0$

Step 2.2.4

Cancel the common factor of $6$ .

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Step 2.2.4.1

Move the leading negative in $- \frac{5 w}{6}$ into the numerator.

$3 w^{2} - 4 + 6 (\frac{- 5 w}{6}) = 0$

Step 2.2.4.2

Cancel the common factor.

$3 w^{2} - 4 + 6 (\frac{- 5 w}{6}) = 0$

Step 2.2.4.3

Rewrite the expression.

$3 w^{2} - 4 - 5 w = 0$

Step 2.3

Move $- 4$ .

$3 w^{2} - 5 w - 4 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = - 5$ , and $c = - 4$ into the quadratic formula and solve for $w$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (3 \cdot - 4)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 5$ to the power of $2$ .

$w = \frac{5 \pm \sqrt{25 - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$w = \frac{5 \pm \sqrt{25 - 12 \cdot - 4}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 4$ .

$w = \frac{5 \pm \sqrt{25 + 48}}{2 \cdot 3}$

Step 5.1.3

Add $25$ and $48$ .

$w = \frac{5 \pm \sqrt{73}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$w = \frac{5 \pm \sqrt{73}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$w = \frac{5 \pm \sqrt{73}}{6}$

Decimal Form:

$w = 2.25733395 \dots, - 0.59066729 \dots$