Problem 1 - Algebra 1

Question

Define the roots of a quadratic equation.

Step-by-Step Solution

Answer

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are the values of \( x \) that satisfy the equation, found using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Step 1: Understanding the Quadratic Equation

A quadratic equation looks like this: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are known numbers and \( x \) is the variable. Also, \( a \) must not be zero.

Step 2: Understanding the Roots

The roots of a quadratic equation are the values of \( x \) that satisfy the equation. They also represent where the graph of the equation crosses the x-axis.

Step 3: Using the Quadratic Formula to Find Roots

The quadratic formula is: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Using this formula, you can find the roots of any quadratic equation. Plug \( a \), \( b \), and \( c \) from your equation into the quadratic formula to find \( x \). Note that depending upon the value under the square root in the formula, you can have: two real roots (if \( b^2 - 4ac > 0 \)), one real root (if \( b^2 - 4ac = 0 \)), or two complex roots (if \( b^2 - 4ac < 0 \)).

Key Concepts

Roots of Equations

The roots of an equation are crucial for finding where a function crosses the x-axis. In the context of quadratic equations, the roots refer to the values of the variable, usually represented as \(x\), that make the equation equal to zero.

A quadratic equation is typically in the form \(ax^2 + bx + c = 0\).

• Here, \(a\), \(b\), and \(c\) are constants.
• \(x\) is the variable whose values are what we are solving for, known as the roots.

To determine the roots, we solve this equation, which essentially finds the points where the parabola described by the equation intersects the x-axis. These roots can tell us a lot about the nature of the curve and are pivotal for graphing and understanding the equation's behavior.

Quadratic Formula

To solve a quadratic equation, we use a specific tool called the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula can be used to find the roots of any quadratic equation.

The process involves:

• Identifying the coefficients \(a\), \(b\), and \(c\) in your equation \(ax^2 + bx + c = 0\).
• Plugging these values into the quadratic formula.
• Calculating the expression under the square root, \(b^2 - 4ac\), known as the discriminant.

After computation, you will have two potential values for \(x\), given by the plus-minus symbol (\(\pm\)), which represents the sum and difference of the square root component.

Real and Complex Roots

The discriminant, \(b^2 - 4ac\), plays a vital role in determining the nature of the roots derived from a quadratic equation. Here's how it works:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots. This scenario occurs when the parabola crosses the x-axis at two points.
• If \(b^2 - 4ac = 0\), there is exactly one real root. It means the parabola merely touches the x-axis, known as a repeated or double root.
• If \(b^2 - 4ac < 0\), the roots are complex or imaginary. The graph does not touch or cross the x-axis, indicating the existence of complex roots, often expressed in terms of \(i\), the imaginary unit.

Understanding whether the roots are real or complex not only helps in solving quadratic equations but also in predicting the graph's interaction with the x-axis, informing both theoretical insights and practical applications.