Solve Using the Quadratic Formula 1/3x^2=-1/6x+1/2

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

Step 1

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

Simplify the left side.

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

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

Simplify the right side.

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

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

Move all the expressions to the left side of the equation.

Add $\frac{x}{6}$ to both sides of the equation.

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

Subtract $\frac{1}{2}$ from both sides of the equation.

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

Step 2

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

Apply the distributive property.

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

Simplify.

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $- 1$ .

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

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 $a = 2$ , $b = 1$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (2 \cdot - 3)}}{2 \cdot 2}$

Step 5

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 3}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 3$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot - 3}}{2 \cdot 2}$

Multiply $- 8$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{1 + 24}}{2 \cdot 2}$

Add $1$ and $24$ .

$x = \frac{- 1 \pm \sqrt{25}}{2 \cdot 2}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 1 \pm \sqrt{5^{2}}}{2 \cdot 2}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 1 \pm 5}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm 5}{4}$

Step 6

The final answer is the combination of both solutions.

$x = 1, - \frac{3}{2}$

Solve Using the Quadratic Formula 1/3x^2=-1/6x+1/2

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