Algebra Examples

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

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

Simplify each term.

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Step 1.1.1.1

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

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

Step 1.1.1.2

Combine $\frac{3}{4}$ and $q$ .

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

Step 1.2

Add $\frac{1}{3}$ to both sides of the equation.

$\frac{q^{2}}{4} + \frac{3 q}{4} + \frac{1}{3} = 0$

Step 2

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

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Step 2.1

Apply the distributive property.

$12 (\frac{q^{2}}{4}) + 12 (\frac{3 q}{4}) + 12 (\frac{1}{3}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Factor $4$ out of $12$ .

$4 (3) (\frac{q^{2}}{4}) + 12 (\frac{3 q}{4}) + 12 (\frac{1}{3}) = 0$

Step 2.2.1.2

Cancel the common factor.

$4 \cdot (3 (\frac{q^{2}}{4})) + 12 (\frac{3 q}{4}) + 12 (\frac{1}{3}) = 0$

Step 2.2.1.3

Rewrite the expression.

$3 q^{2} + 12 (\frac{3 q}{4}) + 12 (\frac{1}{3}) = 0$

Step 2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.1

Factor $4$ out of $12$ .

$3 q^{2} + 4 (3) (\frac{3 q}{4}) + 12 (\frac{1}{3}) = 0$

Step 2.2.2.2

Cancel the common factor.

$3 q^{2} + 4 \cdot (3 (\frac{3 q}{4})) + 12 (\frac{1}{3}) = 0$

Step 2.2.2.3

Rewrite the expression.

$3 q^{2} + 3 (3 q) + 12 (\frac{1}{3}) = 0$

Step 2.2.3

Multiply $3$ by $3$ .

$3 q^{2} + 9 q + 12 (\frac{1}{3}) = 0$

Step 2.2.4

Cancel the common factor of $3$ .

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Step 2.2.4.1

Factor $3$ out of $12$ .

$3 q^{2} + 9 q + 3 (4) (\frac{1}{3}) = 0$

Step 2.2.4.2

Cancel the common factor.

$3 q^{2} + 9 q + 3 \cdot (4 (\frac{1}{3})) = 0$

Step 2.2.4.3

Rewrite the expression.

$3 q^{2} + 9 q + 4 = 0$

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 $a = 3$ , $b = 9$ , and $c = 4$ into the quadratic formula and solve for $q$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (3 \cdot 4)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $9$ to the power of $2$ .

$q = \frac{- 9 \pm \sqrt{81 - 4 \cdot 3 \cdot 4}}{2 \cdot 3}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$q = \frac{- 9 \pm \sqrt{81 - 12 \cdot 4}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $4$ .

$q = \frac{- 9 \pm \sqrt{81 - 48}}{2 \cdot 3}$

Step 5.1.3

Subtract $48$ from $81$ .

$q = \frac{- 9 \pm \sqrt{33}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$q = \frac{- 9 \pm \sqrt{33}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$q = \frac{- 9 \pm \sqrt{33}}{6}$

Decimal Form:

$q = - 0.54257289 \dots, - 2.45742710 \dots$