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Solve using the quadratic formula.\newline8k2+7k−1=08k^2 + 7k - 1 = 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.\newline8k2+7k−1=08k^2 + 7k - 1 = 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 8k2+7k−1=08k^2 + 7k - 1 = 0. Comparing 8k2+7k−1=08k^2 + 7k - 1 = 0 with the standard form ax2+bx+c=0ax^2 + bx + c = 0, we find: a=8a = 8 b=7b = 7 c=−1c = -1
  2. Write quadratic formula: Write down the quadratic formula, which is 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=−(7)±(7)2−4(8)(−1)2(8)k = \frac{{-(7) \pm \sqrt{{(7)^2 - 4(8)(-1)}}}}{{2(8)}}
  4. Simplify discriminant: Simplify the expression under the square root (the discriminant). (7)2−4(8)(−1)=49+32=81\sqrt{(7)^2 - 4(8)(-1)} = \sqrt{49 + 32} = \sqrt{81}
  5. Calculate possible values: Calculate the two possible values for kk.k=−7±8116k = \frac{{-7 \pm \sqrt{81}}}{{16}}k=−7±916k = \frac{{-7 \pm 9}}{{16}}
  6. Find solutions: Find the two solutions by performing the addition and subtraction.\newlineFirst solution:\newlinek=(−7+9)/16k = (-7 + 9) / 16\newlinek=2/16k = 2 / 16\newlinek=1/8k = 1/8\newlineSecond solution:\newlinek=(−7−9)/16k = (-7 - 9) / 16\newlinek=−16/16k = -16 / 16\newlinek=−1k = -1
  7. Simplify solutions: Simplify the solutions and, if necessary, round to the nearest hundredth.\newlineFirst solution is already in simplest form:\newlinek=18k = \frac{1}{8}\newlineSecond solution is an integer:\newlinek=−1k = -1\newlineNo rounding is necessary as both solutions are exact.

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