Verified Answer :

  • Identify coefficients \(a=4\), \(b=24\), \(c=11\) for the quadratic function \(f(x) = 4x^2 + 24x + 11\).
  • Calculate the discriminant \(\Delta = 24^2 - 4(4)(11) = 400\).
  • Use the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\) to find the zeros \(x = -0.5\) and \(x = -5.5\).
  • The correct option is \(x = -5.5\).
  • Final Answer: \(\boxed{-5.5}\)

Explanation

  1. Problem Analysis and Identification of Coefficients

We are tasked with finding a zero of the quadratic function \(f(x) = 4x^2 + 24x + 11\). A zero of a function is a value of \(x\) for which \(f(x) = 0\). This leads to the quadratic equation \(4x^2 + 24x + 11 = 0\), which is in the standard form \(ax^2 + bx + c = 0\), where \(a = 4\), \(b = 24\), and \(c = 11\).

  1. Calculating the Discriminant and Applying the Quadratic Formula

To find the roots of the equation, we apply the quadratic formula, which is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
First, we calculate the discriminant \(\Delta = b^2 - 4ac\). Substituting our coefficients:
\[\Delta = (24)^2 - 4(4)(11)\]
\[\Delta = 576 - 176\]
\[\Delta = 400\]
Since the discriminant is positive, the function has two distinct real roots. The square root of the discriminant is \(\sqrt{400} = 20\).

  1. Solving for the Roots and Selecting the Correct Answer

Now we substitute the values into the quadratic formula to find the two possible values for \(x\):
\[x = \frac{-24 \pm 20}{8}\]
The two solutions are:

  1. \(x_1 = \frac{-24 + 20}{8} = \frac{-4}{8} = -0.5\)
  2. \(x_2 = \frac{-24 - 20}{8} = \frac{-44}{8} = -5.5\)

Comparing these results with the given options (\(x = -9.25\), \(x = -5.5\), \(x = 0.5\), and \(x = 3.25\)), we can see that \(x = -5.5\) is one of the zeros.

Examples

Quadratic equations are used in physics to model the trajectory of objects. For example, if you throw a ball into the air, its height \(h\) at time \(t\) can be modeled by a quadratic equation such as \(h(t) = -16t^2 + v_0t + h_0\). Finding the zeros of this equation helps determine when the ball hits the ground, allowing you to predict its travel time and landing position accurately. Understanding how to solve these equations is essential for predicting outcomes in various real-world scenarios.

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