Problem 75 - Algebra and Trigonometry

Question

Complex Conjugate Roots Suppose that the equation \(a x^{2}+b x+c=0\) has real coefficients and complex roots. Why must the roots be complex conjugates of each other? (Think about how you would find the roots using the quadratic formula.)

Step-by-Step Solution

Answer

The roots are complex conjugates because they have the same real part but opposite imaginary parts when the discriminant is negative.

Step 1: Solve using the Quadratic Formula

The roots of a quadratic equation \(ax^2 + bx + c = 0\) are given by the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\] Here, the discriminant \(b^2 - 4ac\) determines the nature of the roots.

Step 2: Check Discriminant for Complex Roots

For the roots to be complex, the discriminant \(b^2 - 4ac\) must be negative. This means \(\sqrt{b^2 - 4ac}\) will involve the imaginary unit \(i\), giving us \(x = \frac{-b \pm i\sqrt{|b^2 - 4ac|}}{2a}\).

Step 3: Observe Conjugate Pairs

The expression for the roots with a negative discriminant becomes \(x = \frac{-b + i\sqrt{|b^2 - 4ac|}}{2a}\) and \(x = \frac{-b - i\sqrt{|b^2 - 4ac|}}{2a}\). These two solutions are complex conjugates, as they have the same real part, \(-\frac{b}{2a}\), and imaginary parts that are negatives of each other.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to find the roots of a quadratic equation. This formula is written as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]The expression inside the square root, \( b^2 - 4ac \), is crucial because it influences the nature of the roots.

• If \( b^2 - 4ac \) is positive, the roots are real and distinct.
• If \( b^2 - 4ac \) is zero, the roots are real and equal (or repeated).
• If \( b^2 - 4ac \) is negative, the roots will involve imaginary numbers, resulting in complex roots.

Understanding how to apply this formula not only helps us find the roots effectively, but also guides us in predicting their nature.

Discriminant

The discriminant, denoted by \( b^2 - 4ac \), acts as a valuable indicator of the root's properties for any quadratic equation. The value of the discriminant provides insights into whether the roots are real or complex:

• Real and distinct roots: When the discriminant is greater than zero (> 0), the quadratic equation has two separate real number solutions.
• Real and repeated roots: A zero discriminant (= 0) indicates that both roots are the same, often known as repeated or double roots.
• Complex roots: A negative discriminant (< 0) signifies that the roots are not real numbers but are instead part of the complex number set.

For equations with a negative discriminant, the imaginary unit \( i \) appears in the roots, requiring a deeper understanding of complex numbers.

Complex Numbers

Complex numbers come into play when solving quadratic equations with negative discriminants since real numbers are insufficient to express the roots. A complex number is written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \).
Example: The numbers \( 3 + 2i \) and \( 3 - 2i \) are complex conjugates. They have equal real parts (3) but opposite imaginary parts (\( 2i \) and \( -2i \)). This conjugate relationship is vital, especially in quadratic equations with complex roots.
When the discriminant is negative, solutions acquired through the quadratic formula become complex conjugate pairs, such as \( \frac{-b + i\sqrt{|b^2 - 4ac|}}{2a} \) and \( \frac{-b - i\sqrt{|b^2 - 4ac|}}{2a} \). These maintain the symmetry and balance required by equations with real coefficients, confirming that complex numbers and their conjugates are fundamental in such cases.