Problem 29
Question
\(m^{2}=-4\)
Step-by-Step Solution
Verified Answer
The solutions to the equation are m = 2i and m = -2i.
1Step 1: Understand the Equation
The given equation is a quadratic equation in the form of a^{2}=bdetermining that it has no real roots, as the square of a real number cannot be negative.
2Step 2: Rewrite for Solution
To solve this quadratic equation, we need to solve for m such that m^{2}=-4where we can find the square root of both sides
3Step 3: Recognize the Complex Number
We know that the right-hand side of the equation is -4, recognize that the roots will involve imaginary numbers. Recall that √-1=i, then: m = ±2i
Key Concepts
Complex NumbersImaginary UnitSquare Roots
Complex Numbers
Let's dive into the concept of complex numbers. In simple terms, a complex number is just an extension of real numbers. It is written in the form of cmplx{a}{b}, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. When 'b' is not zero, the number is called a complex number. Here, 'a' is the real part, and 'bi' is the imaginary part.
Complex numbers can be very useful because they allow us to work with roots of negative numbers - something that's not possible with just real numbers! They come into play when we solve equations like the given one, where the square of a number turns out to be negative.
Complex numbers can be very useful because they allow us to work with roots of negative numbers - something that's not possible with just real numbers! They come into play when we solve equations like the given one, where the square of a number turns out to be negative.
Imaginary Unit
The imaginary unit 'i' is a fundamental concept when dealing with complex numbers. By definition, 'i' is the square root of -1. Hence, \(i^{2} = -1\). Using 'i', we can express the square roots of negative numbers.
For instance, the square root of -4 can be expressed as \( \pm 2i \), since \( (2i)^{2} = 4 (i^{2}) = 4(-1) = -4 \).
With the help of this magical number 'i', we turn something seemingly impossible into a workable format.
For instance, the square root of -4 can be expressed as \( \pm 2i \), since \( (2i)^{2} = 4 (i^{2}) = 4(-1) = -4 \).
With the help of this magical number 'i', we turn something seemingly impossible into a workable format.
Square Roots
Most students are familiar with square roots of positive numbers. For example, the square root of 4 is 2 because \(2^{2} = 4\). However, when it comes to negative numbers, things become a little tricky. The problem lies with the fact that no real number squared will ever produce a negative number.
This is where our friend the imaginary unit 'i' becomes essential. With 'i', we can define the square roots of negative numbers. For the quadratic equation \(m^{2}=-4\), we take the square root of both sides. This gives us \(m = \pm\sqrt{-4} = \pm 2i\). By understanding these definitions and concepts, solving such equations feels much more straightforward!
This is where our friend the imaginary unit 'i' becomes essential. With 'i', we can define the square roots of negative numbers. For the quadratic equation \(m^{2}=-4\), we take the square root of both sides. This gives us \(m = \pm\sqrt{-4} = \pm 2i\). By understanding these definitions and concepts, solving such equations feels much more straightforward!