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Solve using the quadratic formula.\newline4m2−m−9=04m^2 - m - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinem=m = _____ or m=m = _____

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Q. Solve using the quadratic formula.\newline4m2−m−9=04m^2 - m - 9 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinem=m = _____ or m=m = _____
  1. Quadratic Formula Definition: The quadratic formula is given by m=−b±b2−4ac2am = \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. In this case, a=4a = 4, b=−1b = -1, and c=−9c = -9.
  2. Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. Here, b2=(−1)2=1b^2 = (-1)^2 = 1 and 4ac=4×4×(−9)=−1444ac = 4 \times 4 \times (-9) = -144. So, the discriminant is 1−(−144)=1+144=1451 - (-144) = 1 + 144 = 145.
  3. 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 mm. This gives us m=−(−1)±1452×4.m = \frac{-(-1) \pm \sqrt{145}}{2 \times 4}.
  4. Simplify Equation: Simplify the equation by calculating the numerator for both the plus and minus scenarios. For the plus scenario: m=1+1458m = \frac{1 + \sqrt{145}}{8}. For the minus scenario: m=1−1458m = \frac{1 - \sqrt{145}}{8}.
  5. Calculate Decimal Values: Since the discriminant is a non-perfect square, the square root of 145145 cannot be simplified to a rational number. Therefore, we will leave the square root as is and express mm as two decimals rounded to the nearest hundredth. Calculate the decimal values: m≈(1+12.04)/8m \approx (1 + 12.04) / 8 and m≈(1−12.04)/8m \approx (1 - 12.04) / 8.
  6. Perform Calculations: Perform the calculations for both scenarios. For the plus scenario: m≈(1+12.04)/8≈13.04/8≈1.63m \approx (1 + 12.04) / 8 \approx 13.04 / 8 \approx 1.63. For the minus scenario: m≈(1−12.04)/8≈−11.04/8≈−1.38m \approx (1 - 12.04) / 8 \approx -11.04 / 8 \approx -1.38.

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