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Solve using the quadratic formula.\newline2n2+6n−7=02n^2 + 6n - 7 = 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+6n−7=02n^2 + 6n - 7 = 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 values of aa, bb, cc: Identify the values of aa, bb, and cc in the quadratic equation 2n2+6n−7=02n^2 + 6n - 7 = 0. Compare 2n2+6n−7=02n^2 + 6n - 7 = 0 with the standard form ax2+bx+c=0ax^2 + bx + c = 0. a=2a = 2 bb00 bb11
  2. Substitute values into quadratic formula: Substitute the values of aa, bb, and cc into the quadratic formula n=−b±b2−4ac2an = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. Substitute a=2a = 2, b=6b = 6, and c=−7c = -7 into the quadratic formula. n=−(6)±(6)2−4⋅2⋅(−7)2⋅2n = \frac{-(6) \pm \sqrt{(6)^2 - 4\cdot2\cdot(-7)}}{2\cdot2}
  3. Simplify expression and calculate discriminant: Simplify the expression under the square root and calculate the discriminant.\newline(6)2−4⋅2⋅(−7)\sqrt{(6)^2 - 4\cdot 2\cdot (-7)}\newline= 36+56\sqrt{36 + 56}\newline= 92\sqrt{92}
  4. Continue with quadratic formula: Continue with the quadratic formula using the simplified discriminant.\newlinen=−6±922⋅2n = \frac{-6 \pm \sqrt{92}}{2 \cdot 2}\newlinen=−6±924n = \frac{-6 \pm \sqrt{92}}{4}
  5. Calculate possible solutions for nn: Calculate the two possible solutions for nn.n=−6+924n = \frac{{-6 + \sqrt{92}}}{{4}} or n=−6−924n = \frac{{-6 - \sqrt{92}}}{{4}}
  6. Simplify square root of 9292: Simplify the square root of 9292 to its simplest radical form if possible.\newlineSince 9292 is not a perfect square, we cannot simplify it further in terms of integers or proper fractions. We will use the decimal approximation for the square root.\newline92≈9.59\sqrt{92} \approx 9.59
  7. Substitute approximate value into solutions: Substitute the approximate value of 92\sqrt{92} into the solutions and round to the nearest hundredth.n≈(−6+9.59)/4n \approx (-6 + 9.59) / 4 or n≈(−6−9.59)/4n \approx (-6 - 9.59) / 4n≈3.59/4n \approx 3.59 / 4 or n≈−15.59/4n \approx -15.59 / 4n≈0.90n \approx 0.90 or n≈−3.90n \approx -3.90

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