Problem 79 - Algebra for College Students

Question

Find all real or imaginary solutions to each equation. Use the method of your choice. $$x^{2}=-121$$

Step-by-Step Solution

Answer

The solutions are \(x = 11i\) and \(x = -11i\).

Step 1: Identify the equation

The given equation is \(x^{2} = -121\). Notice that the equation involves a square term set equal to a negative number.

Step 2: Simplify and prepare to solve

To solve the equation for \(x\), we must consider how to handle the square root of a negative number. Recall the concept of imaginary numbers where \(i = \sqrt{-1}\).

Step 3: Rewrite the equation using imaginary unit

Rewrite the equation \(x^{2} = -121\) as \(x^{2} = -1 \times 121\). This can be further written as \(x^{2} = (i \times \text{11})(i \times \text{11})\).

Step 4: Solve for x

Taking the square root of both sides, we obtain \(x = \text{±} \text{11}i\). Thus, the solutions to the equation are \(x = 11i\) and \(x = -11i\).

Key Concepts

Imaginary Numbers

Imaginary numbers might seem a bit tricky at first, but they can be understood with a little practice. In math, the imaginary unit is denoted as \(i\) and defined by the property \(i^2 = -1\). This means that \(i\) is the square root of \(-1\).

When we extend our number system to include imaginary numbers, we can better deal with equations that involve the square roots of negative numbers. For example, consider the term \(\sqrt{-121}\). Normally, we wouldn't be able to find a real number that squares to \(-121\). By using imaginary numbers, we break this negative factor into a part we understand: \(\sqrt{121} = 11\) and \(\sqrt{-1} = i\).

So, \(\sqrt{-121} = \sqrt{121} \times \sqrt{-1} = 11i\). Imaginary numbers help us to expand the toolkit available for solving equations!

Square Root Method

The square root method is an efficient technique for solving quadratic equations of the form \(x^{2} = k\). Here’s how it works:

• First, isolate the square term on one side of the equation. For instance, if you have \(x^{2} = -121\), the square term is already isolated.
• Next, apply the square root to both sides of the equation to solve for \(x\). However, since the equation involves a negative number, you will use the imaginary unit \(i\).

Taking the square root of both sides in \(x^{2} = -121\) results in:

• \(x = \sqrt{-121}\)
• \(x = \sqrt{121 \times -1}\)
• \(x = \sqrt{121} \times \sqrt{-1} \)
• \(x = 11 \times i\)

Thus, the solutions to the equation \(x^{2} = -121\) are \(x = 11i\) and \(x = -11i\). The square root method simplifies the process of resolving such quadratic equations.

Complex Solutions

When solving quadratic equations, sometimes we encounter what are called complex solutions. Complex solutions happen when the equation has no real-number solutions. Instead, the solutions are in the form \(a + bi\) where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

In the equation \(x^{2} = -121\), solving it step by step leads us to the solutions \(x = 11i\) and \(x = -11i\). These are pure imaginary numbers and specific instances of complex numbers where the real part \(a\) is zero.

Complex solutions can seem intimidating, but they just represent numbers that combine a real part and an imaginary part. They are fundamental in many areas of advanced mathematics, engineering, and physics. By understanding and working with imaginary and complex numbers, we expand our ability to solve a wider range of problems.