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Solve using the quadratic formula. 4z^(2)+9z+3=0 Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. z= " or "z=

Solve using the quadratic formula.\newline4z2+9z+3=04z^{2}+9z+3=0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinez=z=\newline or z=z=

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Q. Solve using the quadratic formula.\newline4z2+9z+3=04z^{2}+9z+3=0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinez=z=\newline or z=z=
  1. Identify coefficients: To solve the quadratic equation 4z2+9z+3=04z^2 + 9z + 3 = 0 using the quadratic formula, we first identify the coefficients aa, bb, and cc from the standard form of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. In this case, a=4a = 4, b=9b = 9, and c=3c = 3.
  2. Apply quadratic formula: The quadratic formula is given by z=−b±b2−4ac2az = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. We will substitute the values of aa, bb, and cc into this formula to find the solutions for zz.
  3. Calculate discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. This is 92−4(4)(3)=81−48=339^2 - 4(4)(3) = 81 - 48 = 33.
  4. Substitute values: Now, we can substitute the values into the quadratic formula: z=−9±332⋅4z = \frac{{-9 \pm \sqrt{33}}}{{2 \cdot 4}}.
  5. Simplify expression: Simplify the expression: z=−9±338z = \frac{-9 \pm \sqrt{33}}{8}. This gives us two solutions, one for the plus sign and one for the minus sign.
  6. Solution for plus sign: For the plus sign: z=−9+338z = \frac{-9 + \sqrt{33}}{8}. This cannot be simplified to an integer or a proper fraction, so we will leave it as is or calculate the decimal value if needed.
  7. Solution for minus sign: For the minus sign: z=−9−338z = \frac{{-9 - \sqrt{33}}}{{8}}. Similarly, this cannot be simplified to an integer or a proper fraction, so we will leave it as is or calculate the decimal value if needed.

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