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Solve using the quadratic formula.\newline2n2−7n+4=02n^2 - 7n + 4 = 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.\newline2n2−7n+4=02n^2 - 7n + 4 = 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. Quadratic Formula: The quadratic formula is given by n=−b±b2−4ac2an = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the terms in the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. For the equation 2n2−7n+4=02n^2 - 7n + 4 = 0, a=2a = 2, b=−7b = -7, and c=4c = 4.
  2. Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. Here, it is (−7)2−4(2)(4)(-7)^2 - 4(2)(4).
  3. Find Discriminant: Perform the calculation: 49−32=1749 - 32 = 17. The discriminant is 1717.
  4. Plug Values into Formula: Now, plug the values of aa, bb, and the discriminant into the quadratic formula to find the two possible values for nn.n=−(−7)±17(2×2)n = \frac{-(-7) \pm \sqrt{17}}{(2 \times 2)}
  5. Simplify Equation: Simplify the equation by calculating the numerator for both the plus and minus scenarios.\newlinen=7±174n = \frac{7 \pm \sqrt{17}}{4}
  6. Find Solutions: The two solutions are n=7+174n = \frac{7 + \sqrt{17}}{4} and n=7−174n = \frac{7 - \sqrt{17}}{4}. These cannot be simplified to integers or proper fractions, so we will leave them as is or convert them to decimal form.
  7. Calculate Decimal Solutions: To express the solutions as decimals rounded to the nearest hundredth, calculate each one.\newlineFirst solution: n=7+174≈7+4.124≈11.124≈2.78n = \frac{7 + \sqrt{17}}{4} \approx \frac{7 + 4.12}{4} \approx \frac{11.12}{4} \approx 2.78\newlineSecond solution: n=7−174≈7−4.124≈2.884≈0.72n = \frac{7 - \sqrt{17}}{4} \approx \frac{7 - 4.12}{4} \approx \frac{2.88}{4} \approx 0.72

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