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The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is any equation in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula solves for the values of \(x\) that make the equation true by plugging these constants into the formula:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]In our exercise, the coefficients are \(a = 9\), \(b = 24\), and \(c = 16\). By substituting these values into the quadratic formula, we find that both zeros of the function are \(-\frac{4}{3}\). This means the function touches the x-axis at this point but does not cross it. This result implies that the expression under the square root, known as the discriminant \(b^2 - 4ac\), is zero, confirming one real repeated root at \(x = -\frac{4}{3}\).

A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic function like \(f(x) = 9x^2 + 24x + 16\). The general form of a parabola's equation is \(ax^2 + bx + c\), where:- The coefficient \(a\) determines the openness and direction of the parabola. If \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards. In our equation, since \(a = 9\) (positive), the parabola opens upwards.- The vertex is the highest or lowest point on the parabola, depending on the direction it opens. In this case, it's a minimum point.- The axis of symmetry is a vertical line that runs through the vertex, dividing the parabola into two mirror images. It is given by \(x = -\frac{b}{2a}\).By understanding these features, you can predict the parabola's shape and position on a graph before plotting.

Graphing quadratic functions involves understanding their key features, such as the direction they open and their intercepts. For the function \(f(x) = 9x^2 + 24x + 16\), follow these steps:- **Find the vertex**: The vertex of the parabola, which is also a minimum point because the parabola opens upwards, is calculated by \(x = -\frac{b}{2a}\). Substituting gives \(-\frac{4}{3}\), and when substituted into the function, the value of the function at the vertex is \(0\).- **Plot the vertex and intercepts**: Plot the vertex \((-\frac{4}{3}, 0)\) and the y-intercept at \(y=f(0)=16\). This y-intercept is where the graph crosses the y-axis.- **Sketch the parabola**: Draw a U-shaped curve through these points, with the vertex representing the lowest point.These steps result in an accurate graph that visually represents the function.

Zeros of a function, also known as roots or solutions, are values of \(x\) where the function equals zero. For quadratic functions, these are the x-intercepts of the parabolic graph. In the equation \(f(x) = 9x^2 + 24x + 16\), the zeros were found using the quadratic formula, resulting in \(-\frac{4}{3}\) as a repeated zero.- A zero indicates where the graph touches or crosses the x-axis.- For this function, there is a single repeated zero, meaning the graph only touches the x-axis at \(-\frac{4}{3}\) without crossing it.Understanding zeros is crucial for graph interpretation and solving equations, as they show solutions where the output \(f(x)\) is exactly zero.