Verified Answer :

  • Rearrange the equation \(4x = 32 - x^2\) into standard quadratic form: \(x^2 + 4x - 32 = 0\).
  • Calculate the discriminant \(\Delta = b^2 - 4ac = 4^2 - 4(1)(-32) = 144\).
  • Since \(\Delta > 0\), there are two real roots, meaning the graph crosses the x-axis 2 times.
  • Use the quadratic formula to find the roots: \(x = \frac{-4 \pm 12}{2}\), which gives solutions \(x = 4\) and \(x = -8\).
  • Final Answer: The graph crosses the x-axis \(\boxed{2}\) times, and the solutions are \(\boxed{4, -8}\).

Explanation

  1. Rearranging the Equation to Standard Form

To determine the number of times the graph of the equation \(4x = 32 - x^2\) crosses the x-axis, we first rearrange the equation into the standard form of a quadratic equation, \(ax^2 + bx + c = 0\). Adding \(x^2\) and subtracting 32 from both sides, we get:
\[x^2 + 4x - 32 = 0\]
This is a standard quadratic equation where \(a = 1\), \(b = 4\), and \(c = -32\). The points where the graph of this function crosses the x-axis correspond to the real roots of the equation \(x^2 + 4x - 32 = 0\).

  1. Calculating the Discriminant to Determine the Number of Roots

To find the number of real roots, we calculate the discriminant, denoted by \(\Delta\), using the formula \(\Delta = b^2 - 4ac\).
Substituting the coefficients \(a = 1\), \(b = 4\), and \(c = -32\) into the formula:
\[\Delta = (4)^2 - 4(1)(-32)\]
\[\Delta = 16 + 128 = 144\]
Since the discriminant \(\Delta = 144\) is greater than zero, the quadratic equation has two distinct real roots. Therefore, the graph of the function crosses the x-axis exactly twice.

  1. Finding the Solutions to the Equation

Now, we find the roots of the equation \(x^2 + 4x - 32 = 0\) using the quadratic formula, \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\):
\[x = \frac{-4 \pm \sqrt{144}}{2(1)}\]
\[x = \frac{-4 \pm 12}{2}\]
Calculating the two potential values for \(x\):

  1. For the positive root: \(x = \frac{-4 + 12}{2} = \frac{8}{2} = 4\)
  2. For the negative root: \(x = \frac{-4 - 12}{2} = \frac{-16}{2} = -8\)

Thus, the solutions to the equation are \(4\) and \(-8\). The graph crosses the x-axis 2 times, specifically at \(x = 4\) and \(x = -8\). Choosing from the provided options, we see that the question asks for both the count and the values. Based on our calculations, the graph crosses the x-axis 2 times, and the solutions are 4 and -8.

Examples

In architecture and engineering, finding the roots of a quadratic equation helps determine where a parabolic arch or bridge intersects a horizontal baseline. For example, if a cable follows the path of a parabola defined by \(y = 32 - x^2 - 4x\), knowing where it touches the ground (the x-axis) is essential for anchoring the structure correctly. By solving for \(x^2 + 4x - 32 = 0\), engineers can precisely place support pillars at the points \(x = 4\) and \(x = -8\), ensuring the stability and design integrity of the build.

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