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

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

Step 1

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

Simplify the left side.

Combine $\frac{1}{6}$ and $y^{2}$ .

$\frac{y^{2}}{6} = - y - \frac{3}{2}$

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

Add $y$ to both sides of the equation.

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

Add $\frac{3}{2}$ to both sides of the equation.

$\frac{y^{2}}{6} + y + \frac{3}{2} = 0$

Step 2

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

Apply the distributive property.

$6 (\frac{y^{2}}{6}) + 6 y + 6 (\frac{3}{2}) = 0$

Simplify.

Cancel the common factor of $6$ .

Cancel the common factor.

$6 (\frac{y^{2}}{6}) + 6 y + 6 (\frac{3}{2}) = 0$

Rewrite the expression.

$y^{2} + 6 y + 6 (\frac{3}{2}) = 0$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$y^{2} + 6 y + 2 (3) (\frac{3}{2}) = 0$

Cancel the common factor.

$y^{2} + 6 y + 2 \cdot (3 (\frac{3}{2})) = 0$

Rewrite the expression.

$y^{2} + 6 y + 3 \cdot 3 = 0$

Multiply $3$ by $3$ .

$y^{2} + 6 y + 9 = 0$

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 $a = 1$ , $b = 6$ , and $c = 9$ into the quadratic formula and solve for $y$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 9$ .

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 9}}{2 \cdot 1}$

Multiply $- 4$ by $9$ .

$y = \frac{- 6 \pm \sqrt{36 - 36}}{2 \cdot 1}$

Subtract $36$ from $36$ .

$y = \frac{- 6 \pm \sqrt{0}}{2 \cdot 1}$

Rewrite $0$ as $0^{2}$ .

$y = \frac{- 6 \pm \sqrt{0^{2}}}{2 \cdot 1}$

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

$y = \frac{- 6 \pm 0}{2 \cdot 1}$

$- 6$ plus or minus $0$ is $- 6$ .

$y = \frac{- 6}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{- 6}{2}$

Divide $- 6$ by $2$ .

$y = - 3$

Step 6

The final answer is the combination of both solutions.

$y = - 3$ Double roots

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

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