Question

Write a single that computes the roots of a quadratic equation. Apply "extract method" to it.

Short answer

Use the quadratic formula and extract discriminant as a method for clarity.

Step-by-step solution

Understanding the Quadratic Formula

A quadratic equation is of the form \( ax^2 + bx + c = 0 \). The roots of this equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula helps calculate the solutions of the quadratic equation.

Extract Method for Calculating Discriminant

Extract the computation of the discriminant \( b^2 - 4ac \) into a separate method. This helps in organizing the code and allows reusability. The discriminant is used to determine the nature of the roots (real or complex).

Implement the Root Calculation Method

In the main method, use the extracted discriminant method when applying the quadratic formula. Calculate both possible roots, using \( x_1 = \frac{-b + \sqrt{d}}{2a} \) and \( x_2 = \frac{-b - \sqrt{d}}{2a} \), where \( d \) is the discriminant.

Handling Different Scenarios

Based on the discriminant value, handle different scenarios: if the discriminant is positive, there are two real roots; if zero, there is one real root; and if negative, the roots are complex. Implement condition checks to handle these scenarios when computing roots.

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are typically expressed in the form \( ax^2 + bx + c = 0 \). This formula provides a straightforward way to find the roots, or solutions, of the equation. To use the quadratic formula, you'll want to apply:

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

Here, \( a \), \( b \), and \( c \) represent the coefficients of the quadratic equation. The symbol \( \pm \) indicates that there can be two possible solutions for \( x \), which correspond to the potential roots of the equation.
These roots provide the values of \( x \) where the original quadratic equation equals zero. Using this formula not only offers easy calculations but also reveals important properties of the quadratic such as symmetry around the vertical axis (for graphs of the equation), or the fact that each quadratic equation might have two, one, or no real solutions.

Discriminant

The discriminant is a specific part of the quadratic formula, located under the square root. It plays a crucial role in determining the nature of the roots of a quadratic equation. The discriminant is represented as:

• \( b^2 - 4ac \)

The value of the discriminant tells us whether the roots of the quadratic are real or complex.

• When the discriminant is positive, \( b^2 - 4ac > 0 \), this indicates two distinct real roots.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), there is exactly one real root, often called a repeated or double root.
• A negative discriminant, \( b^2 - 4ac < 0 \), means the equation has two complex roots, which are not real numbers.

Understanding the discriminant helps anticipate the nature of solutions without actually solving the quadratic equation. This can be particularly useful when graphing the quadratic or determining its properties.

Real and Complex Roots

Quadratic equations can have different types of roots based on the value of their discriminant. Knowing whether the roots are real or complex is essential in understanding the solutions of the equation.**Real Roots:**

• These arise when the discriminant \( (b^2 - 4ac) \) is greater than or equal to zero.
• If the discriminant is zero, the quadratic equation has one real root, meaning the graph of the equation touches the x-axis at exactly one point.
• If the discriminant is positive, there are two distinct real roots, indicating that the graph of the equation intersects the x-axis at two points.

**Complex Roots:**

• Complex roots occur when the discriminant is less than zero, \( b^2 - 4ac < 0 \).
• These roots are not on the real number line and are expressed in terms of imaginary numbers, typically including the imaginary unit \( i \), where \( i = \sqrt{-1} \).
• Although they do not represent x-intercepts on a graph, they provide a complete understanding of the equation's behavior in the complex plane.

By considering the nature of roots, you can better predict and analyze the outcomes of a quadratic equation and use this information in practical problem-solving scenarios.