Problem 1 - Algebra 1: Concepts and Skills

Question

What are the roots of a quadratic equation?

Step-by-Step Solution

Answer

The roots of a quadratic equation \{ax^2 + bx + c = 0} are given by the quadratic formula: {x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}}. The nature of the roots (real or complex) depends on the discriminant {b^2 - 4ac}. If it is greater than 0, there are two distinct real roots. If it is equal to 0, there's exactly one real root. If it's less than 0, there are two complex roots.

Step 1: Definition of a Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable x, with a non-zero coefficient of \(x^2\). The general form is \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are coefficients.

Step 2: Understanding the Quadratic formula

The roots of the quadratic equation can be found using the quadratic formula, which is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). Here, \(b^2-4ac\) is called the discriminant, and it determines the number and nature of the roots of the equation.

Step 3: Conditions for roots based on the Discriminant

If the discriminant is greater than 0, the equation has two distinct real roots. If it's equal to 0, there's exactly one real root (also called a repeated root). And if it’s less than 0, then there are two complex roots, which are not real numbers.

Key Concepts

Quadratic Formula

The quadratic formula is a mathematical tool used to find the solutions, also known as roots, of quadratic equations. These equations are in the form \(ax^2 + bx + c = 0\). The variables \(a\), \(b\), and \(c\) are coefficients where \(aeq 0\).
This formula is very helpful because it can provide solutions to any quadratic equation, no matter the coefficients. The formula itself is:

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

With this formula, you can calculate the values of \(x\) that make the equation true. These values are known as the roots. To use the formula, simply plug in the values for \(a\), \(b\), and \(c\) from your equation.
Remember to solve for both the plus \(+\) and minus \(-\) versions in the formula.

Discriminant

The discriminant is a specific part of the quadratic formula that significantly influences the nature of the roots. It is represented by the expression \(b^2 - 4ac\). This value is key because it tells us how many and what kind of roots we can expect from the equation.
The discriminant helps classify the roots as follows:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (often called a repeated real root).
• If \(b^2 - 4ac < 0\), there are two complex roots.

Understanding the discriminant is crucial, as it is the determining factor in not just the number of solutions, but also the type of solutions (real or complex) that exist for a given quadratic equation.

Real Roots

Real roots are solutions to a quadratic equation that are real numbers. They occur when the discriminant (\(b^2 - 4ac\)) is zero or positive.
Here's how real roots are classified:

• If the discriminant is greater than zero (\(b^2 - 4ac > 0\)), there are two different real roots. This means the quadratic graph will intersect the x-axis at two distinct points.
• If the discriminant is exactly zero (\(b^2 - 4ac = 0\)), there is one real root, also known as a repeated or double root. In this case, the graph touches the x-axis at only one point, meaning the vertex of the parabola is on the x-axis.

Real roots are important as they indicate where the graph of a quadratic equation intersects or touches the x-axis.

Complex Roots

Complex roots occur when a quadratic equation does not intersect the x-axis at any point on a real graph. This happens if the discriminant \(b^2 - 4ac\) of the quadratic equation is negative.
Complex roots are not real numbers but a combination of real and imaginary numbers. They usually appear in conjugate pairs, which means if \(u + vi\) is a root, \(u - vi\) will also be a root, where \(i\) is the imaginary unit equal to the square root of -1.
This indicates that the quadratic equation graph sits above or below the x-axis without touching it. Understanding complex roots is crucial when solving equations that do not have real solutions, ensuring that students can handle a variety of quadratic equations.