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Solve using the quadratic formula.\newline3v2+7v+3=03v^2 + 7v + 3 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinev=v = _____ or v=v = _____

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Q. Solve using the quadratic formula.\newline3v2+7v+3=03v^2 + 7v + 3 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinev=v = _____ or v=v = _____
  1. Quadratic Formula Explanation: The quadratic formula is given by v=−b±b2−4ac2av = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the quadratic equation av2+bv+c=0av^2 + bv + c = 0. In this case, a=3a = 3, b=7b = 7, 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. For our equation, the discriminant is (7)2−4(3)(3)(7)^2 - 4(3)(3).
  3. Discriminant Calculation: Perform the calculation: (7)2−4(3)(3)=49−36=13(7)^2 - 4(3)(3) = 49 - 36 = 13.
  4. Plug into Quadratic Formula: Now that we have the discriminant, we can plug it into the quadratic formula along with the values of aa and bb. This gives us two possible solutions for vv: v=−7±132×3v = \frac{-7 \pm \sqrt{13}}{2 \times 3}.
  5. Simplify Solutions: Simplify the solutions: v=−7±136v = \frac{{-7 \pm \sqrt{13}}}{{6}}.
  6. Calculate Decimal Values: Since 13\sqrt{13} cannot be simplified to an integer or a simple fraction, and the problem asks for decimals rounded to the nearest hundredth if necessary, we will calculate the decimal values of the two solutions.
  7. First Solution Calculation: First solution: v=−7+136≈−7+3.616≈−3.396≈−0.57v = \frac{-7 + \sqrt{13}}{6} \approx \frac{-7 + 3.61}{6} \approx \frac{-3.39}{6} \approx -0.57 (rounded to the nearest hundredth).
  8. Second Solution Calculation: Second solution: v=−7−136≈−7−3.616≈−10.616≈−1.77v = \frac{-7 - \sqrt{13}}{6} \approx \frac{-7 - 3.61}{6} \approx \frac{-10.61}{6} \approx -1.77 (rounded to the nearest hundredth).

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