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Solve using the quadratic formula.\newline7r2−3r−4=07r^2 - 3r - 4 = 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.\newline7r2−3r−4=07r^2 - 3r - 4 = 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 ax2+bx+c=0ax^2 + bx + c = 0. In this case, a=7a = 7, b=−3b = -3, and c=−4c = -4.
  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(7)(−4)(-3)^2 - 4(7)(-4).
  3. Discriminant Calculation: Perform the calculation: (−3)2−4(7)(−4)=9−(−112)=9+112=121(-3)^2 - 4(7)(-4) = 9 - (-112) = 9 + 112 = 121.
  4. Apply Quadratic Formula: Since the discriminant is positive, there will be two real solutions. Now, apply the quadratic formula with the calculated discriminant.
  5. Calculate Solutions: Calculate the two solutions using the quadratic formula:\newliner=−(−3)±1212×7r = \frac{-(-3) \pm \sqrt{121}}{2 \times 7}\newliner=3±12114r = \frac{3 \pm \sqrt{121}}{14}
  6. Simplify Square Root: Simplify the square root of 121121, which is 1111: \newliner=(3±11)/14r = (3 \pm 11) / 14
  7. Find Two Solutions: Now, find the two solutions by adding and subtracting 1111 from 33 and then dividing by 1414: First solution: r=(3+11)/14=14/14=1r = (3 + 11) / 14 = 14 / 14 = 1 Second solution: r=(3−11)/14=−8/14=−4/7r = (3 - 11) / 14 = -8 / 14 = -4/7 after simplifying the fraction.

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