Solve using the quadratic formula. $\newline$ $4f^2 - 8f - 2 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $f =$ _ or $f =$ _

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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

4f^2 - 8f - 2 = 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

f =

_____ or

f =

_____

Identify values of $a$ , $b$ , $c$ : Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $4f^2 − 8f − 2 = 0$ . The quadratic equation is in the form $af^2 + bf + c = 0$ . Comparing this with our equation, we get: $a = 4$ $b = -8$ $b$ $0$
Substitute into quadratic formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula. $\newline$ The quadratic formula is $f = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substituting the values we get: $\newline$ $f = \frac{-(-8) \pm \sqrt{(-8)^2 - 4\cdot4\cdot(-2)}}{2\cdot4}$
Simplify terms and calculate discriminant: Simplify the terms inside the square root and calculate the discriminant. $\newline$ The discriminant is $b^2 - 4ac$ . So we have: $\newline$ Discriminant = $(-8)^2 - 4\cdot4\cdot(-2)$ $\newline$ Discriminant = $64 + 32$ $\newline$ Discriminant = $96$
Continue simplifying with discriminant: Continue simplifying the quadratic formula with the calculated discriminant. $\newline$ $f = \frac{8 \pm \sqrt{96}}{8}$ $\newline$ Since $96$ is not a perfect square, we will leave the square root as is for now.
Split solutions with $\pm$ : Simplify the expression further by splitting the $\pm$ into two separate solutions. $f = \frac{8 + \sqrt{96}}{8}$ or $f = \frac{8 - \sqrt{96}}{8}$
Simplify $\sqrt{96}$ : Simplify $\sqrt{96}$ to its simplest radical form. $\sqrt{96}$ can be simplified to $4\sqrt{6}$ because $96 = 16\times6$ and $\sqrt{16} = 4$ . So we have: $f = \frac{8 + 4\sqrt{6}}{8}$ or $f = \frac{8 - 4\sqrt{6}}{8}$
Simplify fractions: Simplify the fractions by dividing both terms in the numerator by $8$ . $f = 1 + \frac{\sqrt{6}}{2}$ or $f = 1 - \frac{\sqrt{6}}{2}$
Round solutions if necessary: If necessary, round the solutions to the nearest hundredth. $\newline$ Since $\sqrt{6}$ is approximately $2.45$ , we can round the solutions as follows: $\newline$ $f \approx 1 + \frac{2.45}{2}$ or $f \approx 1 - \frac{2.45}{2}$ $\newline$ $f \approx 1 + 1.225$ or $f \approx 1 - 1.225$ $\newline$ $f \approx 2.23$ or $f \approx -0.23$