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Solve using the quadratic formula.\newline2n2+8n+5=02n^2 + 8n + 5 = 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+8n+5=02n^2 + 8n + 5 = 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 quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. In this case, a=2a = 2, b=8b = 8, and c=5c = 5.
  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 82−4(2)(5)8^2 - 4(2)(5).
  3. Find Discriminant: Perform the calculation: 64−40=2464 - 40 = 24. The discriminant is 2424.
  4. Plug Values into Formula: Now, plug the values of aa, bb, and the discriminant into the quadratic formula: n=−8±244n = \frac{-8 \pm \sqrt{24}}{4}.
  5. Simplify Square Root: Simplify the square root of the discriminant: 24=(4⋅6)=26.\sqrt{24} = \sqrt{(4\cdot6)} = 2\sqrt{6}.
  6. Substitute Simplified Root: Substitute the simplified square root back into the formula: n=−8±264n = \frac{-8 \pm 2\sqrt{6}}{4}.
  7. Divide by Common Factor: Divide each term by the common factor of 44: n=−2±61n = \frac{-2 \pm \sqrt{6}}{1}.
  8. Two Solutions: Now we have two solutions for nn: n=−2+6n = -2 + \sqrt{6} and n=−2−6n = -2 - \sqrt{6}.
  9. Calculate Decimals: To express the solutions as decimals rounded to the nearest hundredth, calculate each one: n≈−2+2.45≈0.45n \approx -2 + 2.45 \approx 0.45 and n≈−2−2.45≈−4.45n \approx -2 - 2.45 \approx -4.45.

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