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Solve using the quadratic formula.\newline5z2−3z−5=05z^2 - 3z - 5 = 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.\newline5z2−3z−5=05z^2 - 3z - 5 = 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 Explanation: 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 quadratic equation az2+bz+c=0az^2 + bz + c = 0. In this case, a=5a = 5, b=−3b = -3, 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 (−3)2−4(5)(−5)(-3)^2 - 4(5)(-5).
  3. Discriminant Calculation: Perform the calculation: (−3)2−4(5)(−5)=9−(−100)=9+100=109(-3)^2 - 4(5)(-5) = 9 - (-100) = 9 + 100 = 109.
  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=−(−3)±1092×5z = \frac{-(-3) \pm \sqrt{109}}{2 \times 5}
  5. Simplify Equation: Simplify the equation by calculating the numerator for both the positive and negative square root options. z=3±10910z = \frac{3 \pm \sqrt{109}}{10}
  6. Approximate Values: Since 109\sqrt{109} cannot be simplified to an integer or a fraction, we will leave it as a square root. However, we can approximate the values of zz to the nearest hundredth.\newlineFor the positive option: z=(3+109)10≈(3+10.44)10≈13.4410≈1.34z = \frac{(3 + \sqrt{109})}{10} \approx \frac{(3 + 10.44)}{10} \approx \frac{13.44}{10} \approx 1.34\newlineFor the negative option: z=(3−109)10≈(3−10.44)10≈−7.4410≈−0.74z = \frac{(3 - \sqrt{109})}{10} \approx \frac{(3 - 10.44)}{10} \approx \frac{-7.44}{10} \approx -0.74

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