Problem 81 - A Graphical Approach to Precalculus with Limits

Question

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 reverses $$2 i,-2 i$$

Step-by-Step Solution

Answer

The values are \(a = 1\), \(b = 0\), \(c = 4\).

Step 1: Identify the solutions

The problem provides the solutions to the quadratic equation as \(2i\) and \(-2i\). In a quadratic equation, these are the roots.

Step 2: Set up factored form

Since \(2i\) and \(-2i\) are the roots of the quadratic, we can write the equation in factored form: \((x - 2i)(x + 2i) = 0\).

Step 3: Apply zero-product property

According to the zero-product property, if the product of two expressions is zero, then at least one of the expressions must be zero. In our case, \((x - 2i) = 0\) or \((x + 2i) = 0\) will give us the roots \(2i\) and \(-2i\).

Step 4: Expand the factored form

Next, we expand \((x - 2i)(x + 2i) = 0\) using the difference of squares formula:\[ (x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 - (4)(-1) = x^2 + 4 \]

Step 5: Identify values of \(a\), \(b\), and \(c\)

Now that we have \(x^2 + 4 = 0\), we can see that in the standard form \(ax^2 + bx + c = 0\), the coefficients are:- \(a = 1\)- \(b = 0\)- \(c = 4\)

Key Concepts

Zero-Product Property

The zero-product property is a key principle used in solving quadratic equations, among other algebraic equations. It states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is fundamental when dealing with equations set to zero because it allows us to solve for the roots or solutions of the equation by setting each factor equal to zero.

For instance, in the problem, we have

• (x - 2i)(x + 2i) = 0.

Using the zero-product property, we can derive that either (x - 2i) = 0 or (x + 2i) = 0.
These equations are easy to solve:

• First equation yields the root x = 2i
• Second equation gives x = -2i

Thus, the zero-product property provides a reliable way to efficiently break down the product into solvable pieces.

Complex Roots

Complex roots arise when the solutions to a polynomial equation, like a quadratic, involve imaginary numbers. An imaginary number is a multiple of 'i', where 'i' is the square root of -1.
In the exercise, we're dealing with the complex roots

• 2i and -2i

These are not only complex but also conjugates of each other, meaning they have identical real parts and opposite imaginary parts.

Complex roots often appear in pairs when the quadratic equation's discriminant is negative. These pairs, consisting of one root with +i and another with -i, result in no real number solutions for the equation, but instead ones involving imaginary units.
Understanding complex roots is crucial for solving and interpreting equations that stretch beyond the realm of real numbers, opening up a wider field of possibilities in problem-solving applications.

Difference of Squares

The difference of squares is a fundamental algebraic identity used to factor expressions such as

• (x^2 - y^2).

The identity states that

• x^2 - y^2 = (x - y)(x + y).

This identity is particularly useful when simplifying expressions or expanding factored equations back into polynomial form.

In the given problem, the factored form

• (x - 2i)(x + 2i)

is a classic example of a difference of squares pattern. It simplifies to

• x^2 - (2i)^2

and further reduction gives

• x^2 - (-4) which simplifies to x^2 + 4.

Recognizing and applying the difference of squares allows for a streamlined approach to expanding, simplifying, and solving various algebraic equations that initially may seem complex.