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Solve using the quadratic formula.\newline8z2+4z−8=08z^2 + 4z - 8 = 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.\newline8z2+4z−8=08z^2 + 4z - 8 = 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 8z2+4z−8=08z^2 + 4z - 8 = 0, a=8a = 8, b=4b = 4, and c=−8c = -8.
  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 42−4(8)(−8)4^2 - 4(8)(-8).
  3. Discriminant Calculation: Perform the calculation: 16−4(8)(−8)=16+256=27216 - 4(8)(-8) = 16 + 256 = 272.
  4. Insert Values into Formula: Now, insert the values of aa, bb, and the discriminant into the quadratic formula: z=−4±2722×8z = \frac{-4 \pm \sqrt{272}}{2 \times 8}.
  5. Simplify Formula: Simplify the formula: z=−4±27216z = \frac{{-4 \pm \sqrt{272}}}{{16}}.
  6. Find Perfect Square Factors: Since 272\sqrt{272} is not a perfect square, we can simplify it by looking for perfect square factors. The largest perfect square factor of 272272 is 1616, so 272=(16×17)=417\sqrt{272} = \sqrt{(16 \times 17)} = 4\sqrt{17}.
  7. Replace Square Root: Replace 272\sqrt{272} with 4174\sqrt{17} in the formula: z=(−4±417)16z = \frac{(-4 \pm 4\sqrt{17})}{16}.
  8. Factor Out Common Factor: Factor out the common factor of 44 in the numerator: z=4(−1±17)16z = \frac{4(-1 \pm \sqrt{17})}{16}.
  9. Simplify Fraction: Simplify the fraction by dividing both the numerator and the denominator by 44: z=−1±174z = \frac{-1 \pm \sqrt{17}}{4}.
  10. Calculate Solutions: Now we have two solutions for zz, one using the plus sign and one using the minus sign: z=−1+174z = \frac{-1 + \sqrt{17}}{4} or z=−1−174z = \frac{-1 - \sqrt{17}}{4}.
  11. Express as Decimals: To express the solutions as decimals rounded to the nearest hundredth, calculate each one: z≈(−1+4.1231)/4≈3.1231/4≈0.78z \approx (-1 + 4.1231) / 4 \approx 3.1231 / 4 \approx 0.78 and z≈(−1−4.1231)/4≈−5.1231/4≈−1.28z \approx (-1 - 4.1231) / 4 \approx -5.1231 / 4 \approx -1.28.

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