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Solve using the quadratic formula.\newline6z2−2z−7=06z^2 - 2z - 7 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinez=z = _____ or z=z = _____

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Q. Solve using the quadratic formula.\newline6z2−2z−7=06z^2 - 2z - 7 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinez=z = _____ or z=z = _____
  1. Quadratic Formula: The quadratic formula is given by z=−b±b2−4ac2az = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the terms in the quadratic equation az2+bz+c=0az^2 + bz + c = 0. In this case, a=6a = 6, b=−2b = -2, 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. Here, it is (−2)2−4(6)(−7)(-2)^2 - 4(6)(-7).
  3. Perform Calculation: Perform the calculation: 4+168=1724 + 168 = 172.
  4. Insert Values into Formula: Now, insert the values of aa, bb, and the discriminant into the quadratic formula to find the two possible values for zz.z=−(−2)±1722×6z = \frac{-(-2) \pm \sqrt{172}}{2 \times 6}
  5. Simplify Equation: Simplify the equation by calculating the numerator and denominator separately. z=2±17212z = \frac{2 \pm \sqrt{172}}{12}
  6. Factor Out Square Root: Since 172\sqrt{172} is not a perfect square, we can simplify it by factoring out perfect squares. 172\sqrt{172} can be written as (4×43)\sqrt{(4\times43)}, which simplifies to 2432\sqrt{43}.
  7. Substitute Simplified Root: Substitute the simplified square root back into the equation. z=2±24312z = \frac{2 \pm 2\sqrt{43}}{12}
  8. Simplify Fraction: Now, we can simplify the fraction by dividing both terms in the numerator by 22.z=1±436z = \frac{1 \pm \sqrt{43}}{6}
  9. Find Solutions for zz: We have two solutions for zz, which are z=1+436z = \frac{1 + \sqrt{43}}{6} and z=1−436z = \frac{1 - \sqrt{43}}{6}. These cannot be simplified further into integers or fractions, so we can express them as decimals rounded to the nearest hundredth.
  10. Calculate Decimal Values: Calculate the decimal values for both solutions.\newlinez=1+436≈1+6.566≈7.566≈1.26z = \frac{1 + \sqrt{43}}{6} \approx \frac{1 + 6.56}{6} \approx \frac{7.56}{6} \approx 1.26\newlinez=1−436≈1−6.566≈−5.566≈−0.93z = \frac{1 - \sqrt{43}}{6} \approx \frac{1 - 6.56}{6} \approx \frac{-5.56}{6} \approx -0.93

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