Solve Using the Quadratic Formula 3/4s^2+1/2s+1/12=0

$\frac{3}{4} s^{2} + \frac{1}{2} s + \frac{1}{12} = 0$

Step 1

Simplify the left side.

Simplify each term.

Combine $\frac{3}{4}$ and $s^{2}$ .

$\frac{3 s^{2}}{4} + \frac{1}{2} s + \frac{1}{12} = 0$

Combine $\frac{1}{2}$ and $s$ .

$\frac{3 s^{2}}{4} + \frac{s}{2} + \frac{1}{12} = 0$

Step 2

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

Apply the distributive property.

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

Simplify.

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $3$ .

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

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$9 s^{2} + 2 (6) (\frac{s}{2}) + 12 (\frac{1}{12}) = 0$

Cancel the common factor.

$9 s^{2} + 2 \cdot (6 (\frac{s}{2})) + 12 (\frac{1}{12}) = 0$

Rewrite the expression.

$9 s^{2} + 6 s + 12 (\frac{1}{12}) = 0$

Cancel the common factor of $12$ .

Cancel the common factor.

$9 s^{2} + 6 s + 12 (\frac{1}{12}) = 0$

Rewrite the expression.

$9 s^{2} + 6 s + 1 = 0$

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

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

Step 5

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $9$ .

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

Multiply $- 36$ by $1$ .

$s = \frac{- 6 \pm \sqrt{36 - 36}}{2 \cdot 9}$

Subtract $36$ from $36$ .

$s = \frac{- 6 \pm \sqrt{0}}{2 \cdot 9}$

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

$s = \frac{- 6 \pm \sqrt{0^{2}}}{2 \cdot 9}$

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

$s = \frac{- 6 \pm 0}{2 \cdot 9}$

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

$s = \frac{- 6}{2 \cdot 9}$

Multiply $2$ by $9$ .

$s = \frac{- 6}{18}$

Cancel the common factor of $- 6$ and $18$ .

Factor $6$ out of $- 6$ .

$s = \frac{6 (- 1)}{18}$

Cancel the common factors.

Factor $6$ out of $18$ .

$s = \frac{6 \cdot - 1}{6 \cdot 3}$

Cancel the common factor.

$s = \frac{6 \cdot - 1}{6 \cdot 3}$

Rewrite the expression.

$s = \frac{- 1}{3}$

Move the negative in front of the fraction.

$s = - \frac{1}{3}$

Step 6

The final answer is the combination of both solutions.

$s = - \frac{1}{3}$ Double roots

Solve Using the Quadratic Formula 3/4s^2+1/2s+1/12=0

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