Full solution

Q. Solve using the quadratic formula.

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9v^2 + 8v - 6 = 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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v =

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v =

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Identify values: Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $9v^2 + 8v - 6 = 0$ . Compare $9v^2 + 8v - 6 = 0$ with the standard form $ax^2 + bx + c = 0$ . $a = 9$ $b = 8$ $c = -6$
Substitute values: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula $v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substitute $a = 9$ , $b = 8$ , and $c = -6$ into the formula. $v = \frac{-(8) \pm \sqrt{(8)^2 - 4\cdot9\cdot(-6)}}{2\cdot9}$
Simplify expression: Simplify the expression under the square root and calculate its value. $\newline$ $\sqrt{(8)^2 - 4\cdot 9\cdot (-6)}$ $\newline$ = $\sqrt{64 + 216}$ $\newline$ = $\sqrt{280}$
Simplify formula: Simplify the quadratic formula with the calculated square root value. $\newline$ $v = \frac{-8 \pm \sqrt{280}}{18}$ $\newline$ Since $\sqrt{280}$ simplifies to $\sqrt{4\times70}$ which is $2\times\sqrt{70}$ , we can further simplify the expression. $\newline$ $v = \frac{-8 \pm 2\times\sqrt{70}}{18}$
Calculate solutions: Simplify the expression by dividing all terms by $2$ . $\newline$ $v = \frac{-4 \pm \sqrt{70}}{9}$ $\newline$ Now we have two possible solutions for $v$ .
Calculate solutions: Simplify the expression by dividing all terms by $2$ . $\newline$ $v = \frac{-4 \pm \sqrt{70}}{9}$ $\newline$ Now we have two possible solutions for $v$ .Calculate the two possible solutions for $v$ and round them to the nearest hundredth if necessary. $\newline$ First solution: $\newline$ $v = \frac{-4 + \sqrt{70}}{9}$ $\newline$ $v \approx \frac{-4 + 8.37}{9}$ $\newline$ $v \approx \frac{4.37}{9}$ $\newline$ $v \approx 0.49$ $\newline$ Second solution: $\newline$ $v = \frac{-4 - \sqrt{70}}{9}$ $\newline$ $v \approx \frac{-4 - 8.37}{9}$ $\newline$ $v \approx \frac{-12.37}{9}$ $\newline$ $v$ $0$