Question

Without solving, determine the character of the solutions of each equation in the complex number system. $$ 2 x^{2}-4 x+1=0 $$

Short answer

Two distinct real solutions.

Step-by-step solution

Identify a, b, and c

The given quadratic equation is in the form of \(ax^2 + bx + c = 0\). Here, \(a = 2\), \(b = -4\), and \(c = 1\).

Calculate the discriminant

The discriminant \(\Delta\) of a quadratic equation is given by \(\Delta = b^2 - 4ac\).Substituting the values gives \(\Delta = (-4)^2 - 4(2)(1)\).

Simplify the discriminant expression

Simplify the expression: \(\Delta = 16 - 8 = 8\).

Determine the nature of the solutions

Since the discriminant \(\Delta > 0 \), the quadratic equation has two distinct real solutions.

Key concepts

Discriminant

The discriminant is a key concept in determining the nature of the solutions of a quadratic equation. It is given by the formula \(\bigtriangleup = b^2 - 4ac\). Here, \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation of the form \(ax^2 + bx + c = 0\). The discriminant helps us understand the nature of the roots of the equation without actually solving it. By substituting the values of \(a\), \(b\), and \(c\), you can quickly find the value of the discriminant.

Nature of Solutions

The nature of the solutions of a quadratic equation is determined by the value of the discriminant \(\bigtriangleup \)
. Here’s how it works:

• If \(\bigtriangleup < 0\), the quadratic equation has two complex conjugate solutions.
• If \(\bigtriangleup = 0\), there is exactly one real solution, often referred to as a repeated or double root.
• If \(\bigtriangleup > 0\), the equation has two distinct real solutions.

In the given problem, \(\bigtriangleup = 8\), which is greater than zero. Therefore, the quadratic equation \(2x^2 - 4x + 1 = 0\) has two distinct real solutions.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation of the form \(ax^2 + bx + c = 0\). It’s given by: \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
. This formula will provide you with the exact solutions for any quadratic equation using the coefficients \(a\), \(b\), and \(c\). The term inside the square root, \(\bigtriangleup = b^2 - 4ac\), is the discriminant, which we've already discussed helps us to understand the nature of the solutions.

Complex Number System

When dealing with quadratic equations, especially in cases where the discriminant is less than zero, we enter the realm of the complex number system. Complex numbers are numbers that include the imaginary unit \(i\), where \(i^2 = -1\).

• For instance, if the discriminant is negative, the solutions to the quadratic equation will include imaginary numbers.
• These solutions are called complex conjugates and are of the form \(a \pm bi\), where \(a\) and \(b\) are real numbers.

Understanding complex numbers extends our capability to solve any quadratic equation, regardless of the value of the discriminant.