Problem 45 - Algebra 2 and Trigonometry

Question

The difference between the roots of a quadratic equation is 4i. The sum of the roots is 12. Write the equation.

Step-by-Step Solution

Answer

The quadratic equation is \( x^2 - 12x + 40 = 0 \).

Step 1: Understanding the Problem

We are given a quadratic equation with roots whose difference is a purely imaginary number, specifically, \(4i\), and whose sum is a real number, \(12\). This means we have two conditions to satisfy: 1. The difference of the roots \( x_1 - x_2 = 4i \). 2. The sum of the roots \( x_1 + x_2 = 12 \).

Step 2: Express the Roots in Terms of Complex Numbers

Let the roots be \( x_1 = a + bi \) and \( x_2 = c + di \). Given the difference \( x_1 - x_2 = 4i \), it follows that \( (a - c) + (b - d)i = 0 + 4i \). From this, we get two separate equations:- \( a - c = 0 \), implying \( a = c \).- \( b - d = 4 \).

Step 3: Apply the Sum Condition of the Roots

From \( x_1 + x_2 = 12 \), we have:\( (a + bi) + (a + (b-4)i) = 12 \), since \( x_2 \) simplifies to \( a + (b-4)i \). This results in:- \( 2a + 2bi - 4i = 12 \).- Splitting into real and imaginary parts, we get two equations: 1. \( 2a = 12 \) (real part) 2. \( 2bi - 4i = 0 \) (imaginary part).

Step 4: Solve the Real Equation

From the equation \( 2a = 12 \), we divide both sides by 2 to find \( a = 6 \).

Step 5: Solve the Imaginary Equation

From the equation \( 2b - 4 = 0 \), solve for \( b \) by adding 4 to both sides and then dividing by 2:- \( 2b = 4 \)- \( b = 2 \).

Step 6: Find the Roots Using a and b

Since we found \( a = 6 \) and \( b = 2 \), our roots are:- \( x_1 = 6 + 2i \)- \( x_2 = 6 - 2i \) because \( b - 4 = -2 \).

Step 7: Write the Quadratic Equation Using Vieta's Formulas

The general form of a quadratic equation with roots \( x_1 \) and \( x_2 \) is:\( x^2 - (x_1 + x_2)x + x_1x_2 = 0 \).Plug in the sum of the roots (12) and their product:- \( (6 + 2i)(6 - 2i) = 36 - (2i)^2 = 36 - (-4) = 40 \).This gives:\( x^2 - 12x + 40 = 0 \).

Key Concepts

Complex Numbers

When dealing with quadratic equations and their roots, complex numbers can frequently come into play. A complex number is any number of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). Complex numbers allow us to extend the concept of a number line to a number plane, known as the complex plane. This plane has a real axis and an imaginary axis.

• The real part of the complex number is \( a \).
• The imaginary part is \( b \), but it is important to understand that this is \( b \) times the imaginary unit \( i \).

Understanding complex numbers is crucial since many quadratic equations, like the one in the exercise, involve roots that are not real numbers. Working with complex roots involves using both real and imaginary parts to solve equations, as demonstrated in the exercise where the difference between the roots was \(4i\). By interpreting this correctly, we ensure that we account for both dimensions in our calculations.

Roots of Equations

The roots of a quadratic equation are the values of \( x \) that make the equation equal to zero. In other words, they are solutions to the equation. For a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), finding the roots is the same as solving the equation.
In our example exercise, the roots are derived using given conditions: the sum of the roots is real, \(12\), and the difference is imaginary, \(4i\). Using these conditions and expressing the roots in terms of \( a+bi \), the problem is solved by splitting it into simpler equations involving real and imaginary numbers.
Roots can be real or complex and understanding their properties allows us to find various aspects of the equation:

• Real roots are solutions that occur on the real number line.
• Complex roots occur in conjugate pairs, such as \( 6+2i \) and \( 6-2i \), which is a feature of quadratic equations with real coefficients.

These properties of roots help in re-constructing the original quadratic equation, which is crucial for many algebraic solutions.

Vieta's Formulas

Vieta's formulas provide a powerful connection between the roots of a polynomial and its coefficients. For quadratic equations \( ax^2 + bx + c = 0 \), these formulas express the sum and the product of the roots in terms of \( b \) and \( c \). Specifically:

• The sum of the roots \( x_1 \) and \( x_2 \) is given by \( x_1 + x_2 = -\frac{b}{a} \).
• The product of the roots is given by \( x_1 x_2 = \frac{c}{a} \).

These relationships hold true even when the roots are complex numbers, making them versatile tools in algebra. Applying Vieta's formulas to our exercise, we utilized the known sum \( x_1 + x_2 = 12 \) and found the product \( x_1 x_2 = 40 \) from the complex numbers obtained as roots. By substituting these values into the general equation \( x^2 - (x_1 + x_2)x + x_1 x_2 = 0 \), we reconstructed the quadratic equation \( x^2 - 12x + 40 = 0 \). Understanding and applying Vieta's formulas is essential for efficiently solving quadratic equations and unveiling their underlying characteristics.