Problem 79 - A Graphical Approach to College Algebra

Question

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$2 i,-2 i$$

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 + 4 = 0\) with coefficients \(a=1\), \(b=0\), \(c=4\).

Step 1: Understand the Zero-Product Property

To determine the quadratic equation given the solutions, use the reverse zero-product property. If a quadratic equation has roots \(x_1\) and \(x_2\), then it can be expressed as \((x-x_1)(x-x_2)=0\). For this problem, the roots are \(2i\) and \(-2i\).

Step 2: Set Up the Equation with Given Roots

Place the given roots into the equation from Step 1. Thus, the expression is \((x-2i)(x+2i)=0\).

Step 3: Apply the Difference of Squares

Recognize that \((x-2i)(x+2i)\) is a difference of squares. Therefore, \((x-2i)(x+2i) = x^2 - (2i)^2\).

Step 4: Calculate the Squares

Compute \((2i)^2\). Since \(i^2 = -1\), then \((2i)^2 = 4i^2 = 4(-1) = -4\).

Step 5: Formulate the Quadratic Equation

Substitute \(-4\) back into the equation from Step 3. This gives the quadratic equation \(x^2 - (-4) = x^2 + 4 = 0\).

Step 6: Identify Coefficients

Compare the equation \(x^2 + 4 = 0\) with the standard quadratic form \(ax^2 + bx + c = 0\). This yields \(a=1\), \(b=0\), and \(c=4\).

Key Concepts

Zero-Product Property

The Zero-Product Property is a fundamental concept in algebra used to solve quadratic equations. It states that if the product of two factors is zero, then at least one of the factors must also be zero. For instance, if \(a \times b = 0\), either \(a = 0\) or \(b = 0\) (or both). This property is especially helpful when you are trying to find the roots, or solutions, of a quadratic equation.

• To apply this in reverse, like in our exercise, when given rooots \(2i\) and \(-2i\), you use these to form the factors of the quadratic equation: \( (x - 2i)(x + 2i) = 0 \).
• The goal is always to express the quadratic equation in the form where a product equals zero, allowing you to work backwards to identify coefficients from provided solutions.

Understanding this property can significantly simplify solving quadratic equations and is foundational in algebra.

Complex Numbers

Complex numbers introduce the idea of imaginary numbers, which extend our standard numerical system. Numbers like \(2i\) and \(-2i\) are complex because they include \(i\), the imaginary unit.

• The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\), meaning \(i^2 = -1\).
• Numbers that combine real numbers and imaginary numbers are called complex numbers (e.g., \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part).
• In the exercise, the roots \(2i\) and \(-2i\) are pure imaginary numbers, as they lack a real part.

Incorporating complex numbers into solving quadratic equations allows us to find solutions even when the discriminant is negative, expanding the range of solvable equations beyond real numbers.

Difference of Squares

The Difference of Squares is a math trick used when you have two squares being subtracted from each other, such as \(a^2 - b^2\).

• The key formula here is \(a^2 - b^2 = (a - b)(a + b)\).
• In our exercise, this concept simplifies the expression \((x - 2i)(x + 2i)\) into \(x^2 - (2i)^2\).

Application Process:

• Identify the squares in the expression: \((x - ai)(x + ai)\) turns into \(x^2 - a^2i^2\).
• Calculate \((2i)^2\). Keep in mind that since \(i^2 = -1\), \( (2i)^2 = -4\).

This simplification helps transform the quadratic into a standard form, essential for identifying \(a, b,\) and \(c\) in the quadratic equation. It illustrates how recognizing patterns can make problem-solving more efficient.