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A quadratic equation is like a treasure chest, and the quadratic formula is the key to unlock it. The quadratic formula is used to find the zeros or roots of any quadratic equation of the form \[ax^2 + bx + c = 0.\] The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\] For our function, \(f(x) = 9x^2 + 24x + 16\), the coefficients are:

• \(a = 9\)
• \(b = 24\)
• \(c = 16\)

With these values substituted into the quadratic formula, the equation simplifies, revealing the roots. Since the discriminant \(b^2 - 4ac = 0\), this indicates there is a **double root.** This means both roots are the same, which was calculated as\[-\frac{4}{3}.\] Whether dealing with real distinct, real repeated, or complex roots, the quadratic formula is a reliable tool for finding the zeros of quadratic equations.

The vertex of a parabola is a crucial feature. It is where the curve reaches its highest or lowest point. For an upward-opening parabola, like our function, the vertex represents the minimum point. The position of the vertex is determined using \(-\frac{b}{2a}\), which for our equation is \(-\frac{24}{18} = -\frac{4}{3}.\)After finding the x-coordinate of the vertex, substitute back into the original function to determine the y-coordinate:\[f(-\frac{4}{3}) = 9(-\frac{4}{3})^2 + 24(-\frac{4}{3}) + 16.\]Calculating this gives us \(0\), confirming that the minimum value of the function occurs right there at \((-\frac{4}{3}, 0).\) This vertex at \((-\frac{4}{3}, 0)\) means that our graph touches the x-axis at this point and opens upwards as a U-shape.

Zeros of a quadratic equation show where the graph intersects the x-axis. Also called roots, they can be found through the quadratic formula. As we analyzed before, the discriminant \(b^2 - 4ac\) plays a vital role here.

• If the discriminant is greater than zero, expect two distinct real roots.
• If it's exactly zero, you have a double root like we do here, which is \(-\frac{4}{3}.\)
• If the discriminant is less than zero, the roots are complex or imaginary.

In our exercise, the computed root \(-\frac{4}{3}\) is special because it's a double root. This means the parabola just touches the x-axis at one point, creating a unique situation where touching does not mean crossing.

When sketching the graph of a quadratic function like \(f(x) = 9x^2 + 24x + 16\), follow these essentials:

• **Identify the direction**: It is straightforward since the coefficient \(a = 9\) is positive, the parabola opens upwards.
• **Find the vertex**: We've found this at \((-\frac{4}{3}, 0)\). This is the point where the graph touches the x-axis and doesn't dip below.
• **Locate the zeros**: Our zero was calculated to be \(-\frac{4}{3}\), same as the vertex x-coordinate, indicating a double root.

Using these critical points, draw a smooth U-shaped curve that opens upwards. It should touch the x-axis at the single point \(-\frac{4}{3}\), making it clear that the graph has a gentle minimum there. This gives a visual representation of all the mathematical work done.