Solve using the quadratic formula. $\newline$ $9g^2 - 8g - 3 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $g =$ _ or $g =$ _

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Algebra 1

Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

9g^2 - 8g - 3 = 0

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

\newline

g =

_____ or

g =

_____

Identify values of $a$ , $b$ , $c$ : Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $9g^2 − 8g − 3 = 0$ . By comparing the equation to the standard form $ax^2 + bx + c = 0$ , we find: $a = 9$ $b = -8$ $b$ $0$
Substitute values into formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula to find $g$ . The quadratic formula is $g = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . So we have: $g = \frac{-(-8) \pm \sqrt{(-8)^2 - 4\cdot9\cdot(-3)}}{2\cdot9}$
Simplify expression and constants: Simplify the expression under the square root and the constants outside the square root. $\newline$ Calculate the discriminant $b^2 - 4ac$ : $\newline$ Discriminant = $(-8)^2 - 4\cdot 9\cdot (-3)$ $\newline$ Discriminant = $64 + 108$ $\newline$ Discriminant = $172$ $\newline$ Now, simplify the constants: $\newline$ $g = \frac{8 \pm \sqrt{172}}{18}$
Simplify square root: Simplify the square root of the discriminant, if possible. $\newline$ Since $172$ is not a perfect square, we cannot simplify the square root further. Therefore, we have: $\newline$ $g = (8 \pm \sqrt{172}) / 18$
Identify possible values for g: Identify the two possible values for g. We have two solutions for g, corresponding to the ' $\pm$ ' in the quadratic formula: $g = \frac{8 + \sqrt{172}}{18}$ or $g = \frac{8 - \sqrt{172}}{18}$
Approximate values of $g$ : Simplify the fractions or round the values of $g$ to the nearest hundredth, if necessary. $\newline$ First, let's approximate $\sqrt{172}$ to the nearest hundredth: $\newline$ $\sqrt{172} \approx 13.11$ $\newline$ Now, calculate the approximate values of $g$ : $\newline$ $g \approx (8 + 13.11) / 18$ or $g \approx (8 - 13.11) / 18$ $\newline$ $g \approx 21.11 / 18$ or $g \approx -5.11 / 18$ $\newline$ $g \approx 1.17$ or $g$ $0$