Problem 29 - Introductory Algebra for College Students

Question

Solve each quadratic equation using the quadratic formula. $$x^{2}-12 x+40=0$$

Step-by-Step Solution

Answer

The roots of the equation \(x^{2} - 12x + 40 = 0\) are 6 + 2i and 6 - 2i.

Step 1: Identify Coefficients

From the given quadratic equation \(x^{2} - 12x + 40 = 0\), we can identify coefficients as \(a = 1\), \(b = -12\), and \(c = 40\).

Step 2: Calculate The Discriminant

The discriminant \(D\) is calculated as \(D = b^{2} - 4ac = (-12)^{2} - 4*1*40 = 144 - 160 = -16\)

Step 3: Calculate The Roots

Since the discriminant is negative, the roots of the equation are complex and are given by \(x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{-16}}{2} = 6 \pm 2i\).

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is fundamental in algebra. A quadratic equation typically looks like ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

There are several methods to solve these equations, including factoring, completing the square, using graphs, and employing the quadratic formula. The most reliable method, which works when others might fail, is the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). This formula provides solutions by inserting the values of a, b, and c into the formula.

For example, consider the equation x^2 - 12x + 40 = 0. We can identify a = 1, b = -12, and c = 40. Using the quadratic formula, we can find the values of x that make the equation true, which are known as the roots or solutions of the equation.

Discriminant of a Quadratic

The discriminant of a quadratic equation is a key concept that tells us about the nature of the roots without actually solving the equation. It is represented by D and is part of the quadratic formula, found under the square root: D = b^2 - 4ac.

Interpreting the Discriminant:

• If D > 0, the quadratic has two distinct real roots.
• If D = 0, it has exactly one real root, also known as a repeated or double root.
• If D < 0, the quadratic has two complex roots.

For example, in the equation x^2 - 12x + 40 = 0, we calculated the discriminant to be -16, which is less than zero. This tells us that the equation will have complex roots, indicating why the quadratic formula leads to an expression with i, the imaginary unit.

Complex Numbers

Complex numbers are an extension of the real number system and are fundamental in mathematics, especially when dealing with quadratic equations with a negative discriminant.

A complex number is composed of a real part and an imaginary part and is typically written as a + bi, where a is the real part, b is the imaginary part, and i is the square root of -1. When solving quadratic equations with a negative discriminant, the quadratic formula introduces the square root of a negative number, leading to an answer involving complex numbers.

For instance, with the discriminant -16 from our example, we find that the roots are 6 ± 2i, demonstrating the presence of a complex number since the square root of -16 is 4i, where i represents the imaginary unit. These solutions are not real numbers but are valid within the system of complex numbers, which greatly expands our capability to solve various types of equations.