Problem 4 - College Algebra

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; it determines the number and type of solutions based on its value.

Step 1: Understanding the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It is used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). The expression under the radical sign \( b^2 - 4ac \) is called the discriminant.

Step 2: Identifying the Role of the Discriminant

The discriminant \( b^2 - 4ac \) helps determine the nature of the solutions to the quadratic equation. Depending on its value, the type and number of solutions vary.

Step 3: Analyzing the Discriminant's Value

There are three cases based on the value of the discriminant:1. **Positive Discriminant (> 0):** There are two distinct real solutions because the square root of a positive number is real and produces two values when \( \pm \) is considered.2. **Zero Discriminant (= 0):** There is exactly one real solution (a double root) because the square root of zero is zero, leading to one solution.3. **Negative Discriminant (< 0):** There are two complex (imaginary) solutions because the square root of a negative number results in imaginary numbers.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool used to find solutions to quadratic equations. These equations take the standard form:

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero.The quadratic formula is:

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

This formula allows you to calculate the values of \( x \) that satisfy the equation.Understanding the quadratic formula involves recognizing its two parts:

• The signs "+" and "-" denote two potential outcomes, indicating two possible solutions for \( x \).
• The expression \( \sqrt{b^2 - 4ac} \) is crucial for finding these solutions, as it impacts the equation's solvability.

Nature of Solutions

The nature of solutions to a quadratic equation is closely tied to the discriminant, represented by \( b^2 - 4ac \) in the quadratic formula.The discriminant determines how many and what type of solutions the equation will have:

• Positive Discriminant (> 0): Two distinct real solutions exist. When the discriminant is positive, it indicates the parabola crosses the x-axis at two points.
• Zero Discriminant (= 0): One real solution (a double root) is present. This means the parabola touches the x-axis at exactly one point.
• Negative Discriminant (< 0): Two complex solutions occur. A negative discriminant signifies the parabola does not intersect the x-axis, leading to solutions in the complex number system.

By evaluating the value of the discriminant, it's possible to predict and understand the number and types of solutions without solving the entire equation.

Complex Solutions

When the discriminant in a quadratic equation is negative, the solutions become complex or imaginary. This happens because the square root of a negative number isn't defined in the realm of real numbers.Instead, this introduces complex numbers, which include an imaginary unit denoted by \( i \), where \( i = \sqrt{-1} \). Consequently, the solutions will be expressed in terms of imaginary numbers:

• For example, if \( b^2 - 4ac = -k \), then \( \sqrt{b^2 - 4ac} = \sqrt{-k} = i\sqrt{k} \).
• This results in two solutions, \( \frac{-b + i\sqrt{k}}{2a} \) and \( \frac{-b - i\sqrt{k}}{2a} \), which are complex conjugates.

It is important to understand that complex solutions reflect purely theoretical intersections of the parabola with a hypothetical axis in the complex plane, rather than actual crossings of the x-axis on a real coordinate plane.