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Solve using the quadratic formula.\newline9z2+5z−7=09z^2 + 5z - 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.\newline9z2+5z−7=09z^2 + 5z - 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. For the equation 9z2+5z−7=09z^2 + 5z - 7 = 0, a=9a = 9, b=5b = 5, 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 52−4(9)(−7)5^2 - 4(9)(-7).
  3. Perform Calculation: Perform the calculation: 52−4(9)(−7)=25+252=2775^2 - 4(9)(-7) = 25 + 252 = 277.
  4. Apply Quadratic Formula: Now, apply the quadratic formula with the calculated discriminant. The solutions for zz will be z=−5±2772×9z = \frac{-5 \pm \sqrt{277}}{2 \times 9}.
  5. Simplify Solutions: Simplify the solutions: z=−5±27718z = \frac{-5 \pm \sqrt{277}}{18}.
  6. Approximate Square Root: Since 277\sqrt{277} cannot be simplified to a nice integer or fraction, we will need to approximate it to the nearest hundredth for the final answers. 277\sqrt{277} is approximately 16.6416.64.
  7. Calculate Possible Values: Now, calculate the two possible values for zz: z=−5+16.6418z = \frac{{-5 + 16.64}}{{18}} and z=−5−16.6418z = \frac{{-5 - 16.64}}{{18}}.
  8. Calculate Possible Values: Now, calculate the two possible values for zz: z=−5+16.6418z = \frac{-5 + 16.64}{18} and z=−5−16.6418z = \frac{-5 - 16.64}{18}.Perform the calculations: z≈11.6418≈0.65z \approx \frac{11.64}{18} \approx 0.65 and z≈−21.6418≈−1.20z \approx \frac{-21.64}{18} \approx -1.20.

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