Problem 72 - Algebra for College Students

Question

Find all real and imaginary solutions to each equation. $$l^{4}-4=0$$

Step-by-Step Solution

Answer

Real: \( \text{±}\text{√2} \), Imaginary: \( \text{±}i\text{√2} \)

Step 1 - Rewrite the Equation

Rewrite the equation in a simpler form: \[ l^4 - 4 = 0 \] Add 4 to both sides to get: \[ l^4 = 4 \]

Step 2 - Solve for l^2

Take the square root of both sides to find \( l^2 \): \[ (l^2)^2 = 4 \] Therefore, \[ l^2 = \text{±}\text{2} \]

Step 3 - Solve for l

Now, take the square root of both positive and negative sides: For \( l^2 = 2 \), \[ l = \text{±}\text{√2} \] For \( l^2 = -2 \), \[ l = \text{±}\text{√{-2}} \] Simplify it to: \[ l = \text{±}i\text{√2} \]

Step 4 - Collect All Solutions

Gather all solutions found: Real solutions: \[ l = \text{±}\text{√2} \] Imaginary solutions: \[ l = \text{±}i\text{√2} \]

Key Concepts

Real Solutions

In this problem, real solutions are the values of the variable that are not involving the imaginary unit 'i'. In simpler terms, real solutions are numbers you can find on the number line.
When we solved the equation \[ l^4 - 4 = 0 \], we found that \[ l^2 = 2 \].

To find the real solutions, we took the square root of both sides which gave us the solutions \[ l = \text{±}\text{√2} \].
This means our real solutions are the positive and negative square roots of 2.

• Positive square root of 2: \[ l = √2 \]
• Negative square root of 2: \[ l = -√2 \]

Real solutions are especially important because they apply to real-world problems and can be measured or observed directly.

Imaginary Solutions

Sometimes, when solving polynomial equations, we encounter solutions that involve the imaginary unit 'i'. The imaginary unit 'i' is defined as the square root of -1.
When we look at the equation \[ l^2 = -2 \], we realize that taking the square root of a negative number will give us imaginary solutions.

This can be simplified using 'i':
\[ l^2 = -2 \] becomes \[ l = \text{±}\text{√{-2}} \]
Since \[ \text{√{-2}} \] can be written as \[ \text{√{-1}} \times \text{√2} \] and \[ \text{√{-1}} = i \], we get:
\[ l = \text{±}i\text{√2} \]
This means our imaginary solutions are:

• \[ l = i√2 \]
• \[ l = -i√2 \]

Imaginary solutions don't have direct physical applications, but they are crucial in complex mathematics and engineering problems.

Square Roots

A square root of a number x is a number y such that \[ y^2 = x \]. Square roots play a significant role in solving polynomial equations. For example, to solve \[ l^4 - 4 = 0 \], we initially solved for \[ l^2 \] by taking the square root.

The square root function has two main outcomes:

• Positive square root
• Negative square root

For \[ l^2 = 2 \], the square roots are \[ √2 \] and \[ -√2 \].
For \[ l^2 = -2 \], the square roots involve the imaginary unit 'i', giving \[ i√2 \] and \[ -i√2 \].
Knowing how to handle square roots is necessary for solving various types of equations, especially when dealing with real and imaginary numbers together.
Understanding both the real and imaginary components allows for a comprehensive solution to polynomial equations.