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Solve using the quadratic formula.\newline2z2+9z+2=02z^2 + 9z + 2 = 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.\newline2z2+9z+2=02z^2 + 9z + 2 = 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 quadratic equation az2+bz+c=0az^2 + bz + c = 0. In this case, a=2a = 2, b=9b = 9, and c=2c = 2.
  2. Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. For our equation, the discriminant is 92−4(2)(2)9^2 - 4(2)(2).
  3. Discriminant Calculation: Perform the calculation: 92−4(2)(2)=81−16=659^2 - 4(2)(2) = 81 - 16 = 65.
  4. Apply Quadratic Formula: Now, apply the discriminant to the quadratic formula: z=−9±652⋅2z = \frac{{-9 \pm \sqrt{65}}}{{2 \cdot 2}}.
  5. Simplify Formula: Simplify the formula by dividing −9-9 and 65\sqrt{65} by 44: z=−94±654z = \frac{-9}{4} \pm \frac{\sqrt{65}}{4}.
  6. Express Solutions: Since 65\sqrt{65} cannot be simplified to an integer or a simple fraction, we will leave it as is. However, we can express the solutions as decimals if necessary. The two solutions are z=−9+654z = \frac{-9 + \sqrt{65}}{4} and z=−9−654z = \frac{-9 - \sqrt{65}}{4}.
  7. Find Decimal Approximations: To find the decimal approximations, calculate each solution: z≈(−9+65)/4≈(−9+8.06)/4≈−0.24z \approx (-9 + \sqrt{65}) / 4 \approx (-9 + 8.06) / 4 \approx -0.24 and z≈(−9−65)/4≈(−9−8.06)/4≈−4.26z \approx (-9 - \sqrt{65}) / 4 \approx (-9 - 8.06) / 4 \approx -4.26 (rounded to the nearest hundredth).

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