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Solve using the quadratic formula.\newline4k2−2k−7=04k^2 - 2k - 7 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinek=k = _____ or k=k = _____

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Q. Solve using the quadratic formula.\newline4k2−2k−7=04k^2 - 2k - 7 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinek=k = _____ or k=k = _____
  1. Identify coefficients: Identify the coefficients aa, bb, and cc in the quadratic equation 4k2−2k−7=04k^2 − 2k − 7 = 0.a=4a = 4, b=−2b = -2, c=−7c = -7
  2. Write quadratic formula: Write down the quadratic formula: k=−b±b2−4ac2ak = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  3. Substitute values: Substitute the values of aa, bb, and cc into the quadratic formula.k=−(−2)±(−2)2−4⋅4⋅(−7)2⋅4k = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 4 \cdot (-7)}}{2 \cdot 4}
  4. Calculate discriminant: Simplify the equation by calculating the discriminant b2−4acb^2 - 4ac.\newlineDiscriminant = (−2)2−4⋅4⋅(−7)(-2)^2 - 4\cdot4\cdot(-7) = 4+112=1164 + 112 = 116
  5. Insert discriminant: Insert the discriminant back into the quadratic formula. k=2±1168k = \frac{2 \pm \sqrt{116}}{8}
  6. Simplify square root: Simplify the square root of the discriminant if possible.\newline116=(4⋅29)=229\sqrt{116} = \sqrt{(4\cdot29)} = 2\sqrt{29}
  7. Substitute square root: Substitute the simplified square root back into the quadratic formula. k=2±2298k = \frac{2 \pm 2\sqrt{29}}{8}
  8. Divide all terms: Simplify the equation by dividing all terms by 22.k=1±294k = \frac{1 \pm \sqrt{29}}{4}
  9. Calculate solutions: Calculate the two possible solutions for kk.k=1+294k = \frac{1 + \sqrt{29}}{4} or k=1−294k = \frac{1 - \sqrt{29}}{4}
  10. Round values: Round the values of kk to the nearest hundredth, if required.k≈(1+5.39)/4k \approx (1 + 5.39) / 4 or k≈(1−5.39)/4k \approx (1 - 5.39) / 4k≈6.39/4k \approx 6.39 / 4 or k≈−4.39/4k \approx -4.39 / 4k≈1.60k \approx 1.60 or k≈−1.10k \approx -1.10

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