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Solve using the quadratic formula.\newline6f2+6f−6=06f^2 + 6f - 6 = 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.\newline6f2+6f−6=06f^2 + 6f - 6 = 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 quadratic equation: Identify the values of aa, bb, and cc in the quadratic equation 6f2+6f−6=06f^2 + 6f - 6 = 0. The quadratic equation is in the form af2+bf+c=0af^2 + bf + c = 0, so by comparison: a=6a = 6 b=6b = 6 c=−6c = -6
  2. Substitute values into formula: Substitute the values of aa, bb, and cc into the quadratic formula f=−b±b2−4ac2af = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. The quadratic formula is f=−b±b2−4ac2af = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, so we substitute: f=−(6)±(6)2−4⋅6⋅(−6)2⋅6f = \frac{-(6) \pm \sqrt{(6)^2 - 4\cdot6\cdot(-6)}}{2\cdot6}
  3. Simplify discriminant: Simplify the expression under the square root (the discriminant).\newlineCalculate the discriminant: (6)2−4⋅6⋅(−6)(6)^2 - 4\cdot6\cdot(-6)\newline= 36+14436 + 144\newline= 180180
  4. Continue simplifying formula: Continue simplifying the quadratic formula with the calculated discriminant. f=−6±18012f = \frac{-6 \pm \sqrt{180}}{12}
  5. Simplify square root: Simplify the square root of the discriminant, if possible. 180\sqrt{180} can be simplified because 180=36×5180 = 36 \times 5, and 3636 is a perfect square. 180=36×5=36×5=6×5\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6 \times \sqrt{5}
  6. Substitute simplified root: Substitute the simplified square root back into the quadratic formula.\newlinef=−6±6×512f = \frac{{-6 \pm 6 \times \sqrt{5}}}{{12}}
  7. Factor out common terms: Simplify the quadratic formula by factoring out common terms. Both terms in the numerator have a common factor of 66. f=6(−1±5)6×2f=−1±52f = \frac{6(-1 \pm \sqrt{5})}{6\times2} f = \frac{-1 \pm \sqrt{5}}{2}
  8. Identify possible values for f: Identify the two possible values for f. \newlinef=−1+52f = \frac{-1 + \sqrt{5}}{2} or f=−1−52f = \frac{-1 - \sqrt{5}}{2}
  9. Round values to nearest hundredth: If necessary, round the values of ff to the nearest hundredth.f \approx (\-1 + 2.24) / 2 or f \approx (\-1 - 2.24) / 2f≈1.24/2f \approx 1.24 / 2 or f \approx \-3.24 / 2f≈0.62f \approx 0.62 or f \approx \-1.62

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