Problem 77 - Beginning and Intermediate Algebra

Question

Solve each equation. (All solutions are nonreal complex numbers.) $$ x^{2}=-12 $$

Step-by-Step Solution

Answer

The solutions are \( x = 2\sqrt{3}i \) and \( x = -2\sqrt{3}i \).

Step 1: Understand the Problem

We need to find the solutions to the equation \(x^{2} = -12\). The solutions are nonreal complex numbers.

Step 2: Isolate the Term with the Variable

The equation \(x^2 = -12\) is already isolated with the term involving \ x \ on one side of the equation.

Step 3: Take the Square Root of Both Sides

To solve for \ x \, take the square root of both sides of the equation. Remember to include the \( \pm \) symbol: \[ x = \pm \sqrt{-12} \]

Step 4: Simplify the Square Root of the Negative Number

Rewrite \ \sqrt{-12} \ as \ \sqrt{12} \cdot i \. Simplify \ \sqrt{12} \ to \ 2\sqrt{3} \. Thus, we have \[ x = \pm 2\sqrt{3}i \]

Key Concepts

complex numbers

A complex number has two parts: a real part and an imaginary part. It's usually written in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. Complex numbers are very useful in various fields of science and engineering. Here's a key concept to remember: the imaginary part is what differentiates complex numbers from real numbers. For example, in the solution \(x = \text{\textpm} 2\text{\textsqrt{3}}i\), \(2\text{\textsqrt{3}}i\) is the imaginary part.

square roots

Square roots are numbers that produce a specified quantity when multiplied by themselves. For instance, the square root of 9, denoted \(\text{\textsqrt{9}}\), is 3, because \(3 \times 3\) equals 9. In the context of complex numbers, you'll often need to deal with the square roots of negative numbers. For example, \(\text{\textsqrt{-12}}\) can't be simplified using just real numbers. Instead, we use complex numbers to handle it. Remember to simplify square roots by breaking them into their prime factors when possible. \br \br For example, \(\text{\textsqrt{12}}\) simplifies to \(2\text{\textsqrt{3}}\).

imaginary unit

In mathematics, the imaginary unit is denoted by \(i\) and is defined as the square root of -1. This is written as \(i = \text{\textsqrt{-1}}\). Using this definition, any square root of a negative number can be expressed with \(i\). For example, \(\text{\textsqrt{-12}}\) becomes \(\text{\textsqrt{12}} \times i\). By simplifying further, we find that \(\text{\textsqrt{-12}}\) equals \(2\text{\textsqrt{3}}i\). It's important to remember that \(i^2\) always equals -1, which helps when multiplying and simplifying expressions involving complex numbers. \br \br Understanding the imaginary unit is key to solving complex quadratic equations.