Full solution

Q. Solve using the quadratic formula.

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6s^2 - 2s - 8 = 0

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Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

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s =

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s =

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Identify values of $a$ , $b$ , $c$ : Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $6s^2 − 2s − 8 = 0$ . The quadratic equation is in the form $as^2 + bs + c = 0$ . Comparing this with our equation, we get: $a = 6$ $b = -2$ $b$ $0$
Substitute into quadratic formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula $s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substitute $a = 6$ , $b = -2$ , and $c = -8$ into the formula. $s = \frac{-(-2) \pm \sqrt{(-2)^2 - 4\cdot6\cdot(-8)}}{2\cdot6}$
Simplify expression and constants: Simplify the expression under the square root and the constants outside the square root. $\newline$ Calculate $(-2)^2$ , $4\times6\times(-8)$ , and $2\times6$ . $\newline$ $(-2)^2 = 4$ $\newline$ $4\times6\times(-8) = -192$ $\newline$ $2\times6 = 12$ $\newline$ Now substitute these values back into the formula. $\newline$ $s = \frac{2 \pm \sqrt{4 + 192}}{12}$
Simplify expression inside square root: Simplify the expression inside the square root and then the square root itself. $\newline$ Calculate $4 + 192$ . $\newline$ $4 + 192 = 196$ $\newline$ Now take the square root of $196$ . $\newline$ $\sqrt{196} = 14$ $\newline$ Now substitute this value back into the formula. $\newline$ $s = (2 \pm 14) / 12$
Calculate two possible solutions: Calculate the two possible solutions for $s$ . $\newline$ First solution: $\newline$ $s = (2 + 14) / 12$ $\newline$ $s = 16 / 12$ $\newline$ $s = 4 / 3$ $\newline$ Second solution: $\newline$ $s = (2 - 14) / 12$ $\newline$ $s = -12 / 12$ $\newline$ $s = -1$
Write final answers: Simplify the fractions and write the final answers. $\newline$ The first solution $\frac{4}{3}$ is already in its simplest form. $\newline$ The second solution $-1$ is an integer. $\newline$ So the final answers are: $\newline$ $s = \frac{4}{3}$ or $s = -1$