1Cademy - Solving $$\frac{1}{2}u^2 + \frac{2}{3}u = \frac{1}{3}$$ Using the Quadratic Formula

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How to Solve a Quadratic Equation Using the Quadratic Formula

Example

Solving $\frac{1}{2}u^2 + \frac{2}{3}u = \frac{1}{3}$ Using the Quadratic Formula

Solve $\frac{1}{2}u^2 + \frac{2}{3}u = \frac{1}{3}$ by first clearing the fractions and then applying the Quadratic Formula. This example demonstrates how to handle a quadratic equation whose coefficients are fractions — the clearing-fractions technique converts it to integer coefficients before the formula is used.

Preliminary step — Clear the fractions. The denominators are $2$ , $3$ , and $3$ , so the LCD is $6$ . Multiply both sides of the equation by $6$ :

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

Simplify each term: $3u^2 + 4u = 2$ .

Get standard form. Subtract $2$ from both sides:

$3u^2 + 4u - 2 = 0$

Step 1 — Identify $a$ , $b$ , $c$ . Here $a = 3$ , $b = 4$ , and $c = -2$ .

Step 2 — Substitute into the Quadratic Formula:

$u = \frac{-4 \pm \sqrt{4^2 - 4(3)(-2)}}{2(3)}$

Step 3 — Simplify. Inside the square root: $16 - (-24) = 16 + 24 = 40$ :

$u = \frac{-4 \pm \sqrt{40}}{6}$

Simplify the radical: $\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$ :

$u = \frac{-4 \pm 2\sqrt{10}}{6}$

Factor out the common factor of $2$ in the numerator: $\frac{2(-2 \pm \sqrt{10})}{6}$ . Cancel the common factor of $2$ :

$u = \frac{-2 \pm \sqrt{10}}{3}$

The two solutions are $u = \frac{-2 + \sqrt{10}}{3}$ and $u = \frac{-2 - \sqrt{10}}{3}$ . When a quadratic equation has fraction coefficients, multiplying every term by the LCD first transforms the equation into one with integer coefficients, making the subsequent application of the Quadratic Formula much simpler. After applying the formula, the numerator may have a common factor with the denominator that should be factored out and canceled.

Updated 2026-04-21

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OpenStax Elementary Algebra

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