Problem 5 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ z^{2}+5=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation \(z^{2} + 5 = 0\) for the variable z. Answer: The two solutions for z are \(z = \sqrt{5}i\) and \(z = -\sqrt{5}i\).

Step 1: Subtract 5 from both sides of the equation

To start, we will first isolate the term \(z^{2}\). To do this, subtract 5 from both sides of the equation. $$ z^{2} + 5 - 5 = 0 - 5 \\ z^{2} = -5 $$

Step 2: Take the square root of both sides

Now that we have isolated the \(z^{2}\) term, we can take the square root of both sides of the equation. Remember that when taking the square root, we'll have two possible solutions for z: one positive and one negative. $$ \sqrt{z^{2}} = \sqrt{-5} \\ z = \pm \sqrt{-5} $$

Step 3: Simplify the expression

Since the square root of a negative number results in an imaginary number, we cannot further simplify the square root. We can however express the imaginary part using the imaginary unit 'i', where \(i = \sqrt{-1}\). Hence, our simplified solution for z is: $$ z = \pm \sqrt{5}i $$ The two solutions for z are: $$ z = \sqrt{5}i \\ z = -\sqrt{5}i $$

Key Concepts

Complex Numbers

Complex numbers are a type of number that expands upon real numbers by including the concept of imaginary numbers. They have the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).
Complex numbers are quite useful in many areas of mathematics, including solving quadratic equations that have no real solutions, like the one we are discussing here. In our equation, the solution took the form of a pure imaginary number, but it is still technically a complex number with the real part \(a = 0\).
Some key properties of complex numbers include:

• They can be added, subtracted, multiplied, and divided just like real numbers, but with special attention to the \(i\) terms.
• The set of all complex numbers is denoted by \(\mathbb{C}\).
• They are often depicted in a complex plane, which helps visualize operations like addition and multiplication.

Complex numbers form an essential part of higher mathematics, providing solutions in cases where real numbers alone fall short.

Imaginary Numbers

Imaginary numbers come into play when we deal with the square roots of negative numbers. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This defining equation sets the stage for imaginary numbers to extend beyond the realm of real numbers, which do not have real square roots for negative values.
In the quadratic equation \(z^2 + 5 = 0\), taking the square root of \(-5\) results in \(z = \pm \sqrt{5}i\).
This demonstrates that:

• Imaginary numbers are crucial when dealing with cases where real solutions do not exist.
• They allow for more comprehensive solutions and understanding in complex situations, such as oscillations and signal processing.

Understanding imaginary numbers is important because they serve as the building blocks for complex numbers, adding a new dimension to mathematical problems that can't be solved by real numbers alone.

Square Roots

Square roots are a mathematical operation that morphs a number back into its base number. For any positive number \(x\), the square root is a number \(y\) such that \(y^2 = x\). When it comes to negative numbers, the concept extends to require the involvement of imaginary numbers.
Within the original problem, we had to calculate \(\sqrt{-5}\) in order to solve for \(z\).
The process included:

• Recognizing that because negative numbers do not have real square roots, the result will be an imaginary number.
• Expressing the square root as \(\sqrt{5}i\), where \(i = \sqrt{-1}\).
• Understanding the result - that the solutions \(\pm \sqrt{5}i\) are valid in the realm of complex numbers, even though they appear unusual when considered within real numbers alone.

By understanding square roots in the context of negative numbers, one can fully appreciate the necessity for imaginary and complex numbers in expanding our capacity to solve a broader array of equations.