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Solve using the quadratic formula.\newline7z2+4z−1=07z^2 + 4z - 1 = 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.\newline7z2+4z−1=07z^2 + 4z - 1 = 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 Definition: 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=7a = 7, b=4b = 4, and c=−1c = -1.
  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(7)(−1)=16+28=444^2 - 4(7)(-1) = 16 + 28 = 44.
  3. Apply Quadratic Formula: Now, apply the quadratic formula with the values of aa, bb, and cc to find the two possible values for zz.z=−4±442×7z = \frac{-4 \pm \sqrt{44}}{2 \times 7}
  4. Simplify Square Root: Simplify the square root of 4444 to 4×11\sqrt{4 \times 11}, which is 2112\sqrt{11}. \newlinez=−4±21114z = \frac{-4 \pm 2\sqrt{11}}{14}
  5. Split Equation: Now, split the equation into two separate equations, one for the addition and one for the subtraction.\newlineFor addition: z=−4+21114z = \frac{-4 + 2\sqrt{11}}{14}\newlineFor subtraction: z=−4−21114z = \frac{-4 - 2\sqrt{11}}{14}
  6. Simplify Fractions: Simplify both fractions by dividing the numerator and the denominator by 22.\newlineFor addition: z=−2+117z = \frac{-2 + \sqrt{11}}{7}\newlineFor subtraction: z=−2−117z = \frac{-2 - \sqrt{11}}{7}
  7. Convert to Decimals: These are the solutions in their simplest radical form. If we need them as decimals, we can approximate 11\sqrt{11} to be about 3.323.32 (rounded to the nearest hundredth).\newlineFor addition: z≈(−2+3.32)/7≈1.32/7≈0.19z \approx (-2 + 3.32) / 7 \approx 1.32 / 7 \approx 0.19\newlineFor subtraction: z≈(−2−3.32)/7≈−5.32/7≈−0.76z \approx (-2 - 3.32) / 7 \approx -5.32 / 7 \approx -0.76

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