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Solve using the quadratic formula.\newline4f2−8f−2=04f^2 - 8f - 2 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinef=f = _____ or f=f = _____

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Q. Solve using the quadratic formula.\newline4f2−8f−2=04f^2 - 8f - 2 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinef=f = _____ or f=f = _____
  1. Identify values of aa, bb, cc: Identify the values of aa, bb, and cc in the quadratic equation 4f2−8f−2=04f^2 − 8f − 2 = 0. The quadratic equation is in the form af2+bf+c=0af^2 + bf + c = 0. Comparing this with our equation, we get: a=4a = 4 b=−8b = -8 bb00
  2. Substitute into quadratic formula: Substitute the values of aa, bb, and cc into the quadratic formula.\newlineThe quadratic formula is f=−b±b2−4ac2af = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Substituting the values we get:\newlinef=−(−8)±(−8)2−4⋅4⋅(−2)2⋅4f = \frac{-(-8) \pm \sqrt{(-8)^2 - 4\cdot4\cdot(-2)}}{2\cdot4}
  3. Simplify terms and calculate discriminant: Simplify the terms inside the square root and calculate the discriminant.\newlineThe discriminant is b2−4acb^2 - 4ac. So we have:\newlineDiscriminant = (−8)2−4⋅4⋅(−2)(-8)^2 - 4\cdot4\cdot(-2)\newlineDiscriminant = 64+3264 + 32\newlineDiscriminant = 9696
  4. Continue simplifying with discriminant: Continue simplifying the quadratic formula with the calculated discriminant.\newlinef=8±968f = \frac{8 \pm \sqrt{96}}{8}\newlineSince 9696 is not a perfect square, we will leave the square root as is for now.
  5. Split solutions with ±\pm: Simplify the expression further by splitting the ±\pm into two separate solutions.f=8+968f = \frac{8 + \sqrt{96}}{8} or f=8−968f = \frac{8 - \sqrt{96}}{8}
  6. Simplify 96\sqrt{96}: Simplify 96\sqrt{96} to its simplest radical form.96\sqrt{96} can be simplified to 464\sqrt{6} because 96=16×696 = 16\times6 and 16=4\sqrt{16} = 4. So we have: f=8+468f = \frac{8 + 4\sqrt{6}}{8} or f=8−468f = \frac{8 - 4\sqrt{6}}{8}
  7. Simplify fractions: Simplify the fractions by dividing both terms in the numerator by 88.f=1+62f = 1 + \frac{\sqrt{6}}{2} or f=1−62f = 1 - \frac{\sqrt{6}}{2}
  8. Round solutions if necessary: If necessary, round the solutions to the nearest hundredth.\newlineSince 6\sqrt{6} is approximately 2.452.45, we can round the solutions as follows:\newlinef≈1+2.452f \approx 1 + \frac{2.45}{2} or f≈1−2.452f \approx 1 - \frac{2.45}{2}\newlinef≈1+1.225f \approx 1 + 1.225 or f≈1−1.225f \approx 1 - 1.225\newlinef≈2.23f \approx 2.23 or f≈−0.23f \approx -0.23

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