Problem 4 - College Algebra with Corequisite Support

Question

In the quadratic formula, what is the name of the expression under the radical sign \(b^{2}-4 a c\), and how does it determine the number of and nature of our solutions?

Step-by-Step Solution

Answer

The expression is called the discriminant and it determines the nature and number of solutions.

Step 1: Identify the expression

In the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), the expression under the radical sign is \(b^2 - 4ac\). This expression is called the **discriminant**.

Step 2: Determine the discriminant's impact on solutions

The discriminant \(b^2 - 4ac\) determines the number and nature of the solutions to the quadratic equation \(ax^2 + bx + c = 0\). Specifically:- If \(b^2 - 4ac > 0\), there are two distinct real solutions.- If \(b^2 - 4ac = 0\), there is exactly one real solution (a repeated root).- If \(b^2 - 4ac < 0\), there are no real solutions, but two complex solutions.

Key Concepts

Understanding the Discriminant

In the quadratic formula, the discriminant is the part of the formula represented as \( b^2 - 4ac \). This value is crucial because it helps us understand what kind of solutions a quadratic equation will have. To fully grasp its importance, it is vital to break it down into simple steps:

• The discriminant is calculated from the equation \( ax^2 + bx + c = 0 \).
• It provides insight into both the number and types of solutions available.

In essence, the discriminant acts like a key that unlocks the mystery of solutions for quadratic equations, guiding us about the reality and types of the roots. It tells us not just how many solutions we have, but also sheds light on whether these solutions are real or complex.

Real Solutions Explained

When dealing with quadratic equations, real solutions are those that can be found on the number line. They are solutions which are not imaginary or complex, and they occur in the following ways:

• If the discriminant \( b^2 - 4ac > 0 \): There are two distinct real solutions. This situation arises when the quadratic equation's graph, a parabola, intersects the x-axis at two points.
• If the discriminant \( b^2 - 4ac = 0 \): There is exactly one real solution, which means the parabola touches the x-axis at its vertex, resulting in a repeated or double root.

In both of these cases, you're dealing with real numbers, and you can visualize or physically solve them on the coordinate plane. It is crucial in many real-world applications where exact or approximate measurements are necessary.

Navigating Complex Solutions

Complex solutions arise when the discriminant \( b^2 - 4ac < 0 \). This suggests that a quadratic equation has no intersections with the x-axis if plotted on a graph, meaning it doesn't have real number solutions. Instead, the solutions are complex numbers, which have both a real part and an imaginary part.

• These solutions are represented in the form \( a \pm bi \), where \( b \) is non-zero.
• The imaginary unit \( i \) is defined as \( \sqrt{-1} \).

Complex solutions are crucial in many fields such as engineering and physics, where they help in analyzing waves or alternating current circuits. Understanding them enhances our capability to solve problems that extend beyond the real number system.