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Solve using the quadratic formula.\newline4d2+5d+1=04d^2 + 5d + 1 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlined=d = _____ or d=d = _____

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Q. Solve using the quadratic formula.\newline4d2+5d+1=04d^2 + 5d + 1 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlined=d = _____ or d=d = _____
  1. Identify values of aa, bb, cc: Identify the values of aa, bb, and cc in the quadratic equation 4d2+5d+1=04d^2 + 5d + 1 = 0.\newlineThe quadratic equation is in the form ad2+bd+c=0ad^2 + bd + c = 0, so by comparison:\newlinea=4a = 4\newlineb=5b = 5\newlinebb00
  2. Substitute values into formula: Substitute the values of aa, bb, and cc into the quadratic formula to find dd. The quadratic formula is d=−b±b2−4ac2ad = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Substituting the values we get: d=−(5)±(5)2−4⋅4⋅12⋅4d = \frac{-(5) \pm \sqrt{(5)^2 - 4\cdot4\cdot1}}{2\cdot4}
  3. Simplify expression and denominator: Simplify the expression under the square root and the denominator.\newlined=−5±25−168d = \frac{-5 \pm \sqrt{25 - 16}}{8}\newlined=−5±98d = \frac{-5 \pm \sqrt{9}}{8}
  4. Simplify square root and find values: Simplify the square root and identify the two possible values for dd.9=3\sqrt{9} = 3d=(−5±38)d = (\frac{-5 \pm 3}{8})So we have two possible solutions:d=(−5+38)d = (\frac{-5 + 3}{8}) or d=(−5−38)d = (\frac{-5 - 3}{8})
  5. Calculate two values for d: Calculate the two values for d. \newlined=−28d = \frac{-2}{8} or d=−88d = \frac{-8}{8}\newlined=−14d = -\frac{1}{4} or d=−1d = -1
  6. Check for rounding: Check if the solutions need to be rounded to the nearest hundredth. Since both solutions are exact values (one is an integer and the other is a proper fraction), no rounding is necessary.

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