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To solve a quadratic equation like \( f(x) = -12x^2 + 11x + 15 \), we can use the quadratic formula. This method helps find the 'zeros' or 'roots' of the quadratic function, which are the values of \( x \) where \( f(x) = 0 \).

The quadratic formula is represented as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In this formula, \( a \), \( b \), and \( c \) are the coefficients of the terms in the quadratic equation \( ax^2 + bx + c = 0 \). By substituting these coefficients into the formula, we can solve for \( x \).

Using this method, it's crucial to calculate the discriminant first, \( b^2 - 4ac \), which will determine the nature of the roots. Positive discriminants indicate two distinct real roots, negative ones imply complex roots, and a zero discriminant reveals a repeated real root. Now, substitute these values and solve the equation to find your answers.

The vertex of a parabola is a key feature, as it represents the highest or lowest point on the graph, depending on the orientation. While our original quadratic is in the standard form, \( ax^2 + bx + c \), the vertex form is expressed as:

• \( y = a(x-h)^2 + k \)

In this format, \( (h, k) \) represents the vertex of the parabola.

To find the vertex using the standard form, we can calculate \( h \) using \(-\frac{b}{2a}\). For \( f(x) = -12x^2 + 11x + 15 \), we found \( h = \frac{11}{24} \). The next step is to find \( k \), the maximum or minimum value of the function when \( x = h \), by substituting back into the equation to get \( f(h) \).

In this specific case, the function reaches a maximum value since the parabola opens downwards (due to \( a = -12 \) being negative). The vertex for this function is approximately \( \left(\frac{11}{24}, 8.52\right) \). Understanding the vertex form is essential, especially for graphing and analyzing the properties of the parabola.

The discriminant in a quadratic equation is derived from the expression under the square root in the quadratic formula, \( \sqrt{b^2 - 4ac} \). This value, \( b^2 - 4ac \), helps determine the nature of the roots of the quadratic equation. A detailed examination of the discriminant can reveal the following:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If it is negative, the roots are complex or imaginary numbers.

For the function \( f(x) = -12x^2 + 11x + 15 \), the discriminant is calculated as \( 11^2 - 4(-12)(15) = 841 \). Since 841 is a positive perfect square, the function has two distinct real roots, making it straightforward to find these values with exact precision. Understanding and using the discriminant is vital for predicting the behavior of quadratic functions.

A parabola is the graph that represents a quadratic function like \( f(x) = ax^2 + bx + c \). The shape of a parabola is crucial as it tells us important aspects of the quadratic equation's behavior.

Here are some key characteristics of a parabola:

• If \( a \) is positive, the parabola opens upwards, indicating a minimum vertex.
• If \( a \) is negative, the parabola opens downwards, indicating a maximum vertex.
• The vertex is the turning point of the parabola.
• The axis of symmetry passes through the vertex, dividing the parabola into two mirror-image halves.

The parabola for our function is oriented downwards as \( a = -12 \), and its graph demonstrates maximum quality at the vertex \( \left(\frac{11}{24}, 8.52\right) \). Understanding the direction and vertex of a parabola helps us sketch it accurately and comprehend the quadratic's overarching story on a coordinate plane.