Solve Using the Quadratic Formula 81s^2+126s+49=0

$81 s^{2} + 126 s + 49 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 81$ , $b = 126$ , and $c = 49$ into the quadratic formula and solve for $s$ .

$\frac{- 126 \pm \sqrt{126^{2} - 4 \cdot (81 \cdot 49)}}{2 \cdot 81}$

Step 3

Simplify.

Simplify the numerator.

Raise $126$ to the power of $2$ .

$s = \frac{- 126 \pm \sqrt{15876 - 4 \cdot 81 \cdot 49}}{2 \cdot 81}$

Multiply $- 4 \cdot 81 \cdot 49$ .

Multiply $- 4$ by $81$ .

$s = \frac{- 126 \pm \sqrt{15876 - 324 \cdot 49}}{2 \cdot 81}$

Multiply $- 324$ by $49$ .

$s = \frac{- 126 \pm \sqrt{15876 - 15876}}{2 \cdot 81}$

Subtract $15876$ from $15876$ .

$s = \frac{- 126 \pm \sqrt{0}}{2 \cdot 81}$

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

$s = \frac{- 126 \pm \sqrt{0^{2}}}{2 \cdot 81}$

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

$s = \frac{- 126 \pm 0}{2 \cdot 81}$

$- 126$ plus or minus $0$ is $- 126$ .

$s = \frac{- 126}{2 \cdot 81}$

Multiply $2$ by $81$ .

$s = \frac{- 126}{162}$

Cancel the common factor of $- 126$ and $162$ .

Factor $18$ out of $- 126$ .

$s = \frac{18 (- 7)}{162}$

Cancel the common factors.

Factor $18$ out of $162$ .

$s = \frac{18 \cdot - 7}{18 \cdot 9}$

Cancel the common factor.

$s = \frac{18 \cdot - 7}{18 \cdot 9}$

Rewrite the expression.

$s = \frac{- 7}{9}$

Move the negative in front of the fraction.

$s = - \frac{7}{9}$

Step 4

The final answer is the combination of both solutions.

$s = - \frac{7}{9}$ Double roots