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Solve using the quadratic formula.\newline4q2−3q−5=04q^2 - 3q - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlineq=q = _____ or q=q = _____

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Q. Solve using the quadratic formula.\newline4q2−3q−5=04q^2 - 3q - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlineq=q = _____ or q=q = _____
  1. Quadratic Formula: The quadratic formula is given by q=−b±b2−4ac2aq = \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=4a = 4, b=−3b = -3, 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 (−3)2−4(4)(−5)(-3)^2 - 4(4)(-5).
  3. Find Discriminant: Perform the calculation: 9−(−80)=9+80=899 - (-80) = 9 + 80 = 89. The discriminant is 8989.
  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 qq.\newlineq=−(−3)±892×4q = \frac{-(-3) \pm \sqrt{89}}{2 \times 4}
  5. Simplify Equation: Simplify the equation by calculating the numerator for both the positive and negative square root. q=3±898q = \frac{3 \pm \sqrt{89}}{8}
  6. Two Solutions for q: Since the square root of 8989 cannot be simplified further, we have two solutions for q, which are:\newlineq = (3+89)/8(3 + \sqrt{89}) / 8 and q = (3−89)/8(3 - \sqrt{89}) / 8
  7. Calculate Decimals: To express the solutions as decimals rounded to the nearest hundredth, calculate each one.\newlineq≈(3+89)/8≈(3+9.434)/8≈12.434/8≈1.55q \approx (3 + \sqrt{89}) / 8 \approx (3 + 9.434) / 8 \approx 12.434 / 8 \approx 1.55\newlineq≈(3−89)/8≈(3−9.434)/8≈−6.434/8≈−0.80q \approx (3 - \sqrt{89}) / 8 \approx (3 - 9.434) / 8 \approx -6.434 / 8 \approx -0.80

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