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Solve using the quadratic formula.\newlinen2−8n+3=0n^2 - 8n + 3 = 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.\newlinen2−8n+3=0n^2 - 8n + 3 = 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 an2+bn+c=0an^2 + bn + c = 0.\newlineHere, a=1a = 1, b=−8b = -8, and c=3c = 3.
  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=−(−8)±(−8)2−4⋅1⋅32⋅1n = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}.
  4. Simplify inside square root: Simplify inside the square root: (−8)2−4⋅1⋅3=64−12(-8)^2 - 4\cdot1\cdot3 = 64 - 12.\newlinen=8±64−122n = \frac{8 \pm \sqrt{64 - 12}}{2}.
  5. Further simplify square root: Further simplify under the square root: 64−12=5264 - 12 = 52. \newlinen=(8±52)/2n = (8 \pm \sqrt{52}) / 2.
  6. Simplify square root of 5252: Simplify the square root of 5252.52\sqrt{52} is not a perfect square, so it remains as 52\sqrt{52}.n=8±522n = \frac{8 \pm \sqrt{52}}{2}.
  7. Divide by 22: Divide both terms in the numerator by 22.\newlinen=4±522n = \frac{4 \pm \sqrt{52}}{2}.
  8. Calculate two possible values: Simplify 52/2\sqrt{52} / 2.\newlineSince 52\sqrt{52} is not a perfect square, we can't simplify this further without a calculator.\newlinen=4±52/2n = 4 \pm \sqrt{52} / 2.
  9. Approximate square root: Calculate the two possible values for nn.\newlineFirst value: n=4+522n = \frac{4 + \sqrt{52}}{2}.\newlineSecond value: n=4−522n = \frac{4 - \sqrt{52}}{2}.
  10. Find approximate solutions: Use a calculator to approximate 52/2\sqrt{52} / 2.\newline52/2≈3.61\sqrt{52} / 2 \approx 3.61.\newlinen=4±3.61n = 4 \pm 3.61.
  11. Find approximate solutions: Use a calculator to approximate 52/2\sqrt{52} / 2.\newline52/2≈3.61\sqrt{52} / 2 \approx 3.61.\newlinen=4±3.61n = 4 \pm 3.61.Find the two approximate solutions for nn.\newlineFirst value: n≈4+3.61=7.61n \approx 4 + 3.61 = 7.61.\newlineSecond value: n≈4−3.61=0.39n \approx 4 - 3.61 = 0.39.

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