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The quadratic formula is a powerful tool for finding the roots of any quadratic equation. A quadratic equation is generally in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The quadratic formula provides the solutions (or roots) of the equation:

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

This formula makes it very convenient to find the roots, as you only need to plug in the values of \( a \), \( b \), and \( c \). The two solutions arise from the plus-minus (\( \pm \)) sign, which means that you will calculate once with addition and once with subtraction.
The quadratic formula is derived from completing the square of a quadratic equation, a method that involves rewriting the equation to make it easier to solve.

The discriminant is a key part of the quadratic formula and plays a crucial role in determining the nature of the roots. It is the expression under the square root in the quadratic formula:

• \( \Delta = b^2 - 4ac \)

The discriminant provides valuable information:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (or a repeated root).
• If \( \Delta < 0 \), the equation has no real roots (the roots are complex).

For the quadratic given in the exercise, the discriminant is calculated as \( 361 \), which is greater than zero. This tells us that the quadratic equation has two distinct real solutions.

Real roots are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These values are the points where the quadratic function crosses or touches the x-axis.

• When the discriminant is positive, it indicates there are two distinct real roots, meaning the quadratic equation intersects the x-axis at two points.
• When the discriminant is zero, there is exactly one real root, which means the parabola touches the x-axis at a single point, also called a double root.
• When the discriminant is negative, there are no real roots, implying the parabola does not intersect the x-axis at all, but might instead hover above or below it.

Using the quadratic formula, we find that the real roots for the provided function \( p(x) = 5x^2 - 21x + 4 \) are \( x = 4 \) and \( x = 0.2 \). These roots are the solutions to the equation, confirming the intersection at these points.

Identifying the coefficients of a quadratic equation is the first step in solving it using the quadratic formula. For any quadratic equation \( ax^2 + bx + c = 0 \), the coefficients \( a \), \( b \), and \( c \) hold pivotal roles:

• \( a \) is the coefficient of \( x^2 \) and determines the parabola's direction (up if \( a > 0 \), down if \( a < 0 \)).
• \( b \) is the coefficient of \( x \) and impacts the parabola's axis of symmetry.
• \( c \) is the constant term and vertically shifts the parabola on the graph.

For the equation \( p(x) = 5x^2 - 21x + 4 \), the coefficients are identified as:
\( a = 5 \), which indicates the parabola opens upwards.
\( b = -21 \), which helps in determining the shape and position of the axis of symmetry.
\( c = 4 \) adjusts the overall position of the parabola on the graph, shifting it up by 4 units. Understanding these coefficients helps in graphically interpreting the nature of the quadratic function and solving it through the quadratic formula.