Problem 82 - Elementary Algebra

Question

Problem: Solve: \(x^{2}=-11\) Incorrect Answer: $$ \begin{aligned} &x^{2}=-11 \\ &x=\sqrt{-11} \text { or } \quad x=-\sqrt{-11} \\ &x=\pm \sqrt{-11} \end{aligned} $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^2 = -11\) are \(x = i\text{√11}\) and \(x = -i\text{√11}\).

Step 1: Understand the Problem

The problem asks to solve the equation \(x^2 = -11\). This equation involves finding the value of \(x\) such that when squared, the result is -11.

Step 2: Recognize the Properties of Squares

Remember that the square of any real number is always non-negative. Therefore, a real number cannot have a negative square. This implies that the given equation \(x^2 = -11\) does not have a solution in the set of real numbers.

Step 3: Introduce Imaginary Numbers

In order to solve this equation, introduce the concept of imaginary numbers. Recall that the imaginary unit \(i\) is defined such that \(i^2 = -1\).

Step 4: Rewrite the Equation

Express the equation using the imaginary unit. Rewrite \(x^2 = -11\) as \(x^2 = 11 \times (-1)\). Replace \(-1\) with \(i^2\), so the equation becomes \(x^2 = 11i^2\).

Step 5: Solve for \(x\)

Take the square root of both sides of the equation, keeping in mind to consider both positive and negative roots. Thus, \(x = \text{±} \text{√(11i^2)}\). Simplify to get \(x = \text{±} i\text{√11}\).

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that give a negative result when squared. The most basic imaginary number is denoted by the symbol \(i\). By definition, \(i\) satisfies the equation \(i^2 = -1\). Therefore, using \(i\), we can express the square root of any negative number as an imaginary number.
For example:
- The square root of -1 is simply \(i\)
- The square root of -9 is \(3i\) because \(9 = 3^2\)
When solving equations where the square of a number equals a negative number, like in our problem \(x^2 = -11\), we can use \(i\) to simplify our solutions.
Here, \(x = \pm i\sqrt{11}\). This indicates that the solutions are imaginary numbers, not real numbers.

Complex Solutions

Complex solutions involve both real and imaginary parts. A complex number is generally written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
To find complex solutions:
1. Identify the real and imaginary components.
2. Combine them into the form \(a + bi\).

In our example, we found that \(x = \pm i\sqrt{11}\). This shows that there’s no real part (here, \(a = 0\)) and the entire solution is imaginary (\

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\text{√4} = 2\) because \(2 \times 2 = 4\).

Selective properties of square roots include:
- Every positive real number has two square roots: one positive and one negative (e.g., \(\text{√4} = 2\) and \(\text{-√4 = -2}\)).
- The square root of zero is zero (\(\text{√0} = 0\)).
- Negative numbers don’t have real square roots, which brings us back to imaginary numbers.

In our quadratic equation \(x^2 = -11\), taking the square root of both sides, we get \(x = \pm \text{√(-11)} = \pm i \text{√11}\). This demonstrates the relationship between negative square roots and imaginary numbers.