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The term quadratic equation refers to a type of polynomial equation of degree two. This means the highest exponent of the variable, commonly denoted as \(x\), is 2. A standard quadratic equation is usually written in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are numbers known as coefficients, and \(a eq 0\). When \(a\) is zero, the equation becomes linear instead.

• The quadratic term: \(ax^2\)
• The linear term: \(bx\)
• The constant term: \(c\)

Quadratic equations can be solved through various methods, such as factoring, completing the square, or using the quadratic formula. Each of these methods offers a different approach to find the solutions of the equation. In the context of our exercise, instead of solving the equation, we determine the nature (number and type) of its solutions using the discriminant.

Real solutions are the outputs for which a quadratic equation equals zero on the real number line. This means they are the values of \(x\) that make the equation true. Depending on the discriminant (\(D\)), a quadratic equation can have:

• Two distinct real solutions (\(D > 0\))
• One real solution, also called a repeated or double root (\(D = 0\))
• No real solutions, instead it has two complex solutions (\(D < 0\))

In our exercise, since \(D > 0\), the quadratic equation has two distinct real solutions. This positive discriminant indicates that the graph of the equation, which is a parabola, cuts the x-axis at two points.

Coefficients are the numbers that multiply the variables in an equation. In the context of quadratic equations, these are crucial elements:

• \(a\): The coefficient of \(x^2\), must be non-zero.
• \(b\): The coefficient of \(x\).
• \(c\): The constant term, which can be positive or negative.

For the given quadratic equation \(x^2 + rx - s = 0\), the coefficients were identified as \(a = 1\), \(b = r\), and \(c = -s\). Understanding these coefficients is essential because they determine the shape and position of the parabola represented by the equation on a graph.

The discriminant formula is a valuable tool in determining the nature of solutions of a quadratic equation without actually solving it. It is represented as \(D = b^2 - 4ac\). Here's how it works:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution.
• If \(D < 0\), there are no real solutions, instead, there are two complex solutions.

In the exercise, by substituting the coefficients \(a = 1\), \(b = r\), and \(c = -s\) into the discriminant formula, we derived \(D = r^2 + 4s\). With \(s > 0\), it follows that \(D > 0\), confirming the presence of two distinct real solutions. The discriminant provides a quick way to assess what kind of roots the quadratic equation will have.