Problem 2 - College Algebra with Corequisite Support

Question

When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form \(a x^{2}+b x+c=0\) we may graph the equation \(y=a x^{2}+b x+c\) and have no zeroes ( \(x\) -intercepts).

Step-by-Step Solution

Answer

A quadratic can have 0, 1, or 2 solutions. No x-intercepts occur if \(b^2 - 4ac < 0\).

Step 1: Understanding the Problem

We are tasked with determining how many solutions a quadratic equation can have and explaining under what conditions it may have no real solutions. The problem specifically asks us to consider the quadratic equation in the form \( ax^2 + bx + c = 0 \) and relate it to its graph \( y = ax^2 + bx + c \).

Step 2: Determine the Number of Solutions for Quadratics

A quadratic equation \( ax^2 + bx + c = 0 \) can have either two real solutions, one real solution, or no real solutions, this is determined by the discriminant \( b^2 - 4ac \). We start by seeking two solutions because the highest degree (2) indicates up to two solutions are possible. Solutions appear where the graph intersects the x-axis.

Step 3: Explain Zero x-intercepts Possibility

If the discriminant \( b^2 - 4ac \) is negative, there are no real roots, meaning the graph \( y = ax^2 + bx + c \) does not intersect the x-axis. Therefore, there are no x-intercepts or zero points, and the quadratic equation has no real solutions. This scenario corresponds to the parabola being entirely above or below the x-axis depending on the leading coefficient \( a \).

Key Concepts

Discriminant

The discriminant is a key concept when solving quadratic equations. It is represented by the expression \( b^2 - 4ac \). To understand its purpose, consider it as a tool that tells us about the nature of the solutions of the quadratic equation \( ax^2 + bx + c = 0 \).
Here's why the discriminant is so important:

• If the discriminant is positive (greater than zero), the quadratic equation has two distinct real solutions.
• If the discriminant equals zero, it means there is exactly one real solution, or more specifically, the two solutions are identical (also known as a "double root").
• If the discriminant is negative, the equation has no real solutions. This situation is explained further in terms of graph intersections later on.

The discriminant essentially tells us how many times, if at all, the parabola associated with the quadratic equation intersects the x-axis, which segues us into understanding real solutions.

Real Solutions

Real solutions to a quadratic equation are the points where the parabola intersects the x-axis. These are the solutions that we can see on a graph as the points where \( y \) (or the output of the equation) equals zero. Finding these solutions gives us the x-values (solutions) for which the equation \( ax^2 + bx + c = 0 \) holds true.
When we calculate the solutions:

• We get **two real solutions** when the discriminant is positive, meaning the parabola crosses the x-axis at two different points.
• We have **one real solution** when the discriminant is zero, indicating the parabola touches the x-axis at just one point.
• We have **no real solutions** when the discriminant is negative, suggesting that the parabola does not touch or intersect the x-axis at all.

Real solutions are crucial because they are essentially the "answers" to the quadratic equation that can be observed directly from the graph of the equation.

X-Intercepts

X-intercepts are where the function \( y = ax^2 + bx + c \) crosses the x-axis. This is essentially when the quadratic equation is "equal to zero."
Understanding x-intercepts is straightforward with these pointers:

• The x-intercepts correspond to the real solutions of the quadratic equation \( ax^2 + bx + c = 0 \).
• If the discriminant is positive, there will be two x-intercepts, since the parabola crosses the x-axis twice.
• If the discriminant is zero, there will be one x-intercept, indicating the vertex of the parabola touches the x-axis.
• With a negative discriminant, there are no x-intercepts, as the parabola remains entirely above or below the x-axis, depending on the leading coefficient \( a \).

In summary, x-intercepts allow us to visually verify where the solutions of a quadratic equation occur on a graph.