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Solve using the quadratic formula.\newline2x2−x−7=02x^2 - x - 7 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____

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Q. Solve using the quadratic formula.\newline2x2−x−7=02x^2 - x - 7 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____
  1. Quadratic Formula Explanation: The quadratic formula is given by x=−b±b2−4ac2ax = \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=2a = 2, b=−1b = -1, and c=−7c = -7.
  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, this is (−1)2−4(2)(−7)(-1)^2 - 4(2)(-7).
  3. Discriminant Calculation: Perform the calculation: 1−(−56)=1+56=571 - (-56) = 1 + 56 = 57. The discriminant is 5757.
  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 xx.x=−(−1)±572×2x = \frac{-(-1) \pm \sqrt{57}}{2 \times 2}x=1±574x = \frac{1 \pm \sqrt{57}}{4}
  5. Find Two Solutions: Since 57\sqrt{57} cannot be simplified into a simpler radical and is not a perfect square, we will leave it as 57\sqrt{57}. The two solutions are:\newlinex=1+574x = \frac{1 + \sqrt{57}}{4} or x=1−574x = \frac{1 - \sqrt{57}}{4}
  6. Calculate Decimal Approximations: To express the solutions as decimals rounded to the nearest hundredth, we calculate each one.\newlinex≈(1+57)/4≈(1+7.55)/4≈8.55/4≈2.14x \approx (1 + \sqrt{57}) / 4 \approx (1 + 7.55) / 4 \approx 8.55 / 4 \approx 2.14\newlinex≈(1−57)/4≈(1−7.55)/4≈−6.55/4≈−1.64x \approx (1 - \sqrt{57}) / 4 \approx (1 - 7.55) / 4 \approx -6.55 / 4 \approx -1.64

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