Problem 60 - Algebra and Trigonometry

Question

Find all solutions of the equation and express them in the form \(a+b i .\) $$ 2 x^{2}+3=2 x $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2} \pm \frac{i \sqrt{5}}{2}\).

Step 1: Write Down the Equation

Start by writing down the given equation: \[ 2x^2 + 3 = 2x \]

Step 2: Move All Terms to One Side

Subtract \(2x\) from both sides to set the equation to zero.\[ 2x^2 - 2x + 3 = 0 \]

Step 3: Identify Coefficients for Quadratic Formula

Here, the equation is in the standard quadratic form \(ax^2 + bx + c = 0\) where \(a = 2\), \(b = -2\), and \(c = 3\).

Step 4: Apply the Quadratic Formula

Use the quadratic formula, \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), to find \(x\).Substitute the coefficients:\[ x = \frac{{-(-2) \pm \sqrt{{(-2)^2 - 4 \times 2 \times 3}}}}{2 \times 2} \]

Step 5: Calculate the Discriminant

Calculate the discriminant, \(b^2 - 4ac\):\[ (-2)^2 - 4 \times 2 \times 3 = 4 - 24 = -20 \]

Step 6: Solve Using the Discriminant

Since the discriminant is negative, the solutions will be complex.Continue with the quadratic formula:\[ x = \frac{2 \pm \sqrt{-20}}{4} \]Simplify to:\[ x = \frac{2 \pm i \sqrt{20}}{4} \]

Step 7: Simplify Square Root of Negative Expression

Rewrite \(\sqrt{-20}\) using \(i\):\[ \sqrt{-20} = i \sqrt{20} = i \sqrt{4 \times 5} = 2i \sqrt{5} \]

Step 8: Final Form of Solutions

Substitute \(2i \sqrt{5}\) back into the formula:\[ x = \frac{2 \pm 2i \sqrt{5}}{4} \]Simplify:\[ x = \frac{1}{2} \pm \frac{i \sqrt{5}}{2} \]

Step 9: Write Solutions in Standard Form

The solutions are:\[ x_1 = \frac{1}{2} + \frac{i \sqrt{5}}{2} \]\[ x_2 = \frac{1}{2} - \frac{i \sqrt{5}}{2} \]These are in the form \(a + bi\).

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of degree two. It typically takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents an unknown variable. In the given exercise, the quadratic equation derived was \(2x^2 - 2x + 3 = 0\).
Quadratics always involve a squared term — among others — making them nonlinear. The solutions to these equations are found where the curve representing the equation intersects the x-axis on a graph. However, these intersections can involve real numbers or, as it is in this case, complex numbers.
Depending on the values of \(a\), \(b\), and \(c\), a quadratic equation can have up to two distinct solutions. Understanding quadratic equations is fundamental, as they lay the groundwork for more complex mathematical topics.

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. It provides a straightforward method to find the roots of any quadratic equation. This formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• \(b^2 - 4ac\) is known as the discriminant.
• The symbol \(\pm\) indicates that there will be two solutions.

In the context of the exercise, using the coefficients \(a = 2\), \(b = -2\), and \(c = 3\), the quadratic formula helps find two solutions for the equation.
The formula simplifies the complex task of factoring or completing the square into a simple plug-and-play process, making it highly practical in educational settings.

Discriminant

The discriminant, represented by \(b^2 - 4ac\), is a key component of the quadratic formula. It determines the nature and number of solutions of a quadratic equation. Here's what the discriminant tells you:

• If it's positive, there are two distinct real solutions.
• If it's zero, there is one real solution.
• If it's negative, as in the exercise \(-20\), the equation has two complex solutions.

In our example, the discriminant was negative, indicating solutions are not real numbers but complex numbers. Understanding the discriminant's sign provides insight into the types of solutions you'll encounter if you solve the equation, which is critically useful in polarized systems or oscillatory scenarios.

Imaginary Numbers

Imaginary numbers arise when we need to take the square root of a negative number. The fundamental unit of imaginary numbers is \(i\), which is defined as \(\sqrt{-1}\). In the quadratic solutions above, the discriminant was negative, resulting in complex solutions involving imaginary numbers.
When we see \(\sqrt{-20}\), it translates to \(i \sqrt{20}\) in terms of imaginary numbers. The \(i\) constitutes the imaginary component, and in our solution, it formed part of the solutions: \(\frac{i \sqrt{5}}{2}\) for both positive and negative cases, seen as the terms \(\frac{1}{2} \pm \frac{i \sqrt{5}}{2}\).
This interplay between real numbers and \(i\) is vital to fully grasp the concept of complex numbers. Imaginary numbers are not 'imaginary' in the sense of non-existent; instead, they have practical applications in electromagnetism, fluid dynamics, and quantum physics.