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Solve using the quadratic formula.\newline3y2−y−5=03y^2 - y - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newliney=y = _____ or y=y = _____

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Q. Solve using the quadratic formula.\newline3y2−y−5=03y^2 - y - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newliney=y = _____ or y=y = _____
  1. Quadratic Formula: The quadratic formula is given by y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients from the quadratic equation ay2+by+c=0ay^2 + by + c = 0. In this case, a=3a = 3, 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 (−1)2−4(3)(−5)(-1)^2 - 4(3)(-5).
  3. Find Discriminant: Perform the calculation: 1−(−60)=1+60=611 - (-60) = 1 + 60 = 61. The discriminant is 6161.
  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 yy.y=−(−1)±612×3y = \frac{-(-1) \pm \sqrt{61}}{2 \times 3}
  5. Simplify Equation: Simplify the equation by calculating the numerator for both the plus and minus scenarios.\newliney=1±616y = \frac{1 \pm \sqrt{61}}{6}
  6. Write Solutions: Since 61\sqrt{61} cannot be simplified to an integer or a simple fraction, we will leave it as is and write the two solutions for yy.\newliney=1+616y = \frac{1 + \sqrt{61}}{6} or y=1−616y = \frac{1 - \sqrt{61}}{6}
  7. Approximate Solutions: If necessary, approximate the solutions to the nearest hundredth. y≈(1+7.81)/6≈8.81/6≈1.47y \approx (1 + 7.81) / 6 \approx 8.81 / 6 \approx 1.47 or y≈(1−7.81)/6≈−6.81/6≈−1.14y \approx (1 - 7.81) / 6 \approx -6.81 / 6 \approx -1.14

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