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Solve using the quadratic formula.\newline2p2+p−5=02p^2 + p - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinep=p = _____ or p=p = _____

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Q. Solve using the quadratic formula.\newline2p2+p−5=02p^2 + p - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinep=p = _____ or p=p = _____
  1. Quadratic Formula: The quadratic formula is given by p=−b±b2−4ac2ap = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients from the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. In this case, a=2a = 2, b=1b = 1, 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 12−4(2)(−5)=1+40=411^2 - 4(2)(-5) = 1 + 40 = 41.
  3. Apply Formula: Now, apply the quadratic formula with the values of aa, bb, and cc:p=−1±412⋅2p = \frac{{-1 \pm \sqrt{41}}}{{2\cdot2}}p=−1±414p = \frac{{-1 \pm \sqrt{41}}}{4}
  4. Two Solutions: Since 41\sqrt{41} cannot be simplified further, we have two solutions for pp: \newlinep=−1+414p = \frac{-1 + \sqrt{41}}{4} or p=−1−414p = \frac{-1 - \sqrt{41}}{4}
  5. Decimal Approximations: These solutions cannot be simplified to integers or proper fractions. They can be approximated as decimals:\newlinep≈(−1+41)/4≈(−1+6.40)/4≈5.40/4≈1.35p \approx (-1 + \sqrt{41}) / 4 \approx (−1 + 6.40) / 4 \approx 5.40 / 4 \approx 1.35 (rounded to the nearest hundredth)\newlinep≈(−1−41)/4≈(−1−6.40)/4≈−7.40/4≈−1.85p \approx (-1 - \sqrt{41}) / 4 \approx (−1 - 6.40) / 4 \approx -7.40 / 4 \approx -1.85 (rounded to the nearest hundredth)

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