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Solve using the quadratic formula.\newline4r2+9r+3=04r^2 + 9r + 3 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newliner=r = _____ or r=r = _____

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Q. Solve using the quadratic formula.\newline4r2+9r+3=04r^2 + 9r + 3 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newliner=r = _____ or r=r = _____
  1. Quadratic Formula: The quadratic formula is given by r=−b±b2−4ac2ar = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the terms in the quadratic equation ar2+br+c=0ar^2 + br + c = 0. In this case, a=4a = 4, b=9b = 9, and c=3c = 3.
  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 92−4(4)(3)9^2 - 4(4)(3).
  3. Find Solutions: Perform the calculation: 81−48=3381 - 48 = 33. The discriminant is 3333.
  4. Simplify Equation: Now, plug the values of aa, bb, and the discriminant into the quadratic formula to find the two possible values for rr.r=−9±332×4r = \frac{-9 \pm \sqrt{33}}{2 \times 4}
  5. Irrational Number Solutions: Simplify the equation by performing the operations: r=−9±338r = \frac{-9 \pm \sqrt{33}}{8}
  6. Two Solutions: Since the discriminant is positive and not a perfect square, the solutions will be irrational numbers. We can leave the square root as is or approximate the values to the nearest hundredth.
  7. Approximate Square Root: The two solutions are: r=−9+338r = \frac{-9 + \sqrt{33}}{8} and r=−9−338r = \frac{-9 - \sqrt{33}}{8}
  8. Calculate Approximate Values: If we want to round to the nearest hundredth, we approximate the square root of 3333 and perform the division:\newline33≈5.74\sqrt{33} \approx 5.74 (rounded to the nearest hundredth)\newliner≈(−9+5.74)/8r \approx (-9 + 5.74) / 8 and r≈(−9−5.74)/8r \approx (-9 - 5.74) / 8
  9. Calculate Approximate Values: If we want to round to the nearest hundredth, we approximate the square root of 3333 and perform the division:\newline33≈5.74\sqrt{33} \approx 5.74 (rounded to the nearest hundredth)\newliner≈(−9+5.74)/8r \approx (-9 + 5.74) / 8 and r≈(−9−5.74)/8r \approx (-9 - 5.74) / 8Calculate the approximate values:\newliner≈(−3.26)/8r \approx (-3.26) / 8 and r≈(−14.74)/8r \approx (-14.74) / 8\newliner≈−0.41r \approx -0.41 and r≈−1.84r \approx -1.84 (rounded to the nearest hundredth)

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