How many times does the graph of $4x = 32 - x^2$ cross the $x$-axis?
What are the solutions to the equation?
The equation provided is \(4x = 32 - x^2\). To determine the points where the graph crosses the \(x\)-axis, we need to find the values of \(x\) that satisfy this equation. We begin by rewriting it in the standard form of a quadratic equation, \(ax^2 + bx + c = 0\), by moving all terms to one side:
\[x^2 + 4x - 32 = 0\]
Here, we can identify the coefficients as \(a = 1\), \(b = 4\), and \(c = -32\).
To find the number of times the graph crosses the \(x\)-axis, we examine the discriminant \(\Delta = b^2 - 4ac\). This value helps determine the number of real roots for the quadratic equation:
\[\Delta = (4)^2 - 4(1)(-32)\]
\[\Delta = 16 + 128 = 144\]
Since \(\Delta = 144 > 0\), there are two distinct real roots, meaning the graph crosses the \(x\)-axis exactly two times.
We use the quadratic formula to find the roots \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\):
\[x = \frac{-4 \pm \sqrt{144}}{2(1)}\]
\[x = \frac{-4 \pm 12}{2}\]
Calculating the two possible values for \(x\):
\(x_1 = \frac{-4 + 12}{2} = \frac{8}{2} = 4\)
\(x_2 = \frac{-4 - 12}{2} = \frac{-16}{2} = -8\)
The solutions to the equation are \(x = 4\) and \(x = -8\). This matches the pair \((4, -8)\) from the provided options.
In engineering and physics, solving quadratic equations is essential for determining the trajectories of projectiles. For example, if you throw a ball, its height \(h\) at a time \(t\) can often be modeled by a quadratic equation. Finding when the ball hits the ground involves finding the roots of that equation, just as we did here to find when the function crosses the \(x\)-axis. Understanding these roots helps in predicting exactly when and where an object will land.