Problem 6 - Algebra for College Students

Question

First use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$2 x^{2}-x+5=0$$

Step-by-Step Solution

Answer

The equation has two nonreal complex solutions: \(x = \frac{1 \pm i\sqrt{39}}{4}\).

Step 1: Identify the coefficients

For the quadratic equation \(2x^2 - x + 5 = 0\), identify the coefficients: \(a = 2\), \(b = -1\), and \(c = 5\).

Step 2: Calculate the discriminant

The discriminant \(\Delta\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). Using the coefficients identified, calculate \(\Delta = (-1)^2 - 4(2)(5) = 1 - 40 = -39\).

Step 3: Classify the nature of the roots

Since the discriminant \(\Delta = -39\) is less than zero, it indicates that the quadratic equation has two nonreal complex solutions.

Step 4: Solve the quadratic equation

Use the quadratic formula to find the roots: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute in the values to get: \[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot 5}}{2 \cdot 2} = \frac{1 \pm \sqrt{-39}}{4}.\]

Step 5: Simplify the solution

The solutions can be simplified: \(x = \frac{1 \pm i\sqrt{39}}{4}\), indicating two nonreal complex solutions. Therefore, the solutions are \(x = \frac{1 + i\sqrt{39}}{4}\) and \(x = \frac{1 - i\sqrt{39}}{4}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. In general, it is of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown variable. The coefficient \( a \) must not be zero because if \( a = 0 \), the equation becomes linear, not quadratic.
A quadratic equation can have different types of solutions, which depend on the value of the discriminant. The discriminant is a part of the quadratic formula and is crucial in determining the nature of the equation's solutions. The nature of the roots of a quadratic equation can be:

• Two distinct real solutions
• One real solution with a multiplicity of two (repeated root)
• Two nonreal complex solutions

These solutions occur based on the sign of the discriminant, which is \( b^2 - 4ac \). Hence, understanding and finding the discriminant is the first step in analyzing any quadratic equation.

Complex Solutions

Complex solutions in the context of quadratic equations refer to solutions that are not real numbers. These occur when the discriminant is less than zero. If the discriminant is negative, the quadratic equation does not intersect the x-axis, meaning it has no real solutions. Instead, the solutions are complex.

Complex numbers come in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\). Because the discriminant is negative in this scenario, the square root in the quadratic formula yields an imaginary number:

• The solutions are symmetric, as they involve both \(+\) and \(-\) versions of the imaginary component.
• Complex solutions always occur in conjugate pairs, e.g., \( \frac{1 + i\sqrt{39}}{4} \) and \( \frac{1 - i\sqrt{39}}{4} \).

Complex solutions are essential to understand when the quadratic equations represent systems beyond simple real-world problems, such as electrical engineering or signal processing.

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula uses the coefficients \(a\), \(b\), and \(c\) to find the roots, or solutions, of the equation.

The quadratic formula is particularly useful because it applies to all types of quadratic equations, whether they have real or complex solutions. Here’s how it works:

• Calculate \(b^2 - 4ac\) to determine the discriminant.
• Plug the coefficients and the discriminant back into the formula.
• The \(\pm\) symbol represents the two possible solutions: one with a plus and another with a minus.

This formula is vital for solving problems where factoring is difficult or impossible, and it provides a straightforward way to find solutions even when they involve complex numbers. In this exercise, where the discriminant is negative, implementing the quadratic formula leads to complex solutions.