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Solve using the quadratic formula.\newline8n2−n−2=08n^2 - n - 2 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinen=n = _____ or n=n = _____

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Q. Solve using the quadratic formula.\newline8n2−n−2=08n^2 - n - 2 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinen=n = _____ or n=n = _____
  1. Identify coefficients: Identify coefficients aa, bb, and cc from the quadratic equation 8n2−n−2=08n^2 - n - 2 = 0.\newlinea=8a = 8, b=−1b = -1, c=−2c = -2.
  2. Write quadratic formula: Write down the quadratic formula: n=−b±b2−4ac2an = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  3. Plug values into formula: Plug the values of aa, bb, and cc into the quadratic formula.n=−(−1)±(−1)2−4(8)(−2)2(8).n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(8)(-2)}}{2(8)}.
  4. Simplify square root: Simplify inside the square root: (−1)2−4(8)(−2)=1+64(-1)^2 - 4(8)(-2) = 1 + 64.
  5. Calculate value inside root: Calculate the value inside the square root: 1+64=651 + 64 = 65.
  6. Simplify equation: Simplify the equation: n=1±6516n = \frac{1 \pm \sqrt{65}}{16}.
  7. Find two solutions: Find the two solutions for nn.\newlineFirst solution: n=1+6516n = \frac{1 + \sqrt{65}}{16}.\newlineSecond solution: n=1−6516n = \frac{1 - \sqrt{65}}{16}.
  8. Calculate decimal values: Calculate the decimal values for both solutions, rounded to the nearest hundredth.\newlineFirst solution: n≈(1+8.06)/16≈9.06/16≈0.57n \approx (1 + 8.06) / 16 \approx 9.06 / 16 \approx 0.57.\newlineSecond solution: n≈(1−8.06)/16≈−7.06/16≈−0.44n \approx (1 - 8.06) / 16 \approx -7.06 / 16 \approx -0.44.

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