Problem 32 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ m^{2}=-9 $$

Step-by-Step Solution

Answer

The solutions for m in the equation are \( m = 3i \) and \( m = -3i \)

Step 1: Adapt equation

To solve the given problem, one must first understand that here the roots will be complex numbers—due to the negative number under the radical. Therefore the equation can be rewritten in a more conventional form: \(m^{2} = -1*9\)

Step 2: Identify square roots

The roots of this equation will be the square root of both sides. However, in this case the square root of both sides yields a complex number, as the square root of a negative number is a complex number. Therefore, one must remember that the square root of \( -1 \) will be 'i', so it will have two solutions: \( \sqrt{9} * i \) and \( -\sqrt{9} * i \)

Step 3: Solve for m

After realizing step 2, it's straightforward to understand that the two solutions to \( m \) will be: \( m = 3i \) and \( m = -3i \)

Key Concepts

Complex Numbers

Complex numbers are a key concept in advanced mathematics, particularly when dealing with quadratic equations. They are an extension of the real number system and include all the real numbers plus an additional element, the imaginary unit denoted by 'i', where \( i^2 = -1 \). A complex number is written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

In our exercise \( m^2 = -9 \), we encounter a situation where the solution involves taking a square root of a negative number. This is not possible within the real numbers, which is why complex numbers come into play. The square roots of -9, being \( \pm3i \), are complex numbers. Here, 'i' is indicative of the fact that we have moved beyond the realm of real numbers into that of complex numbers to find a solution to the given equation.

Radical Expressions

A radical expression is an expression that includes a radical symbol (\( \sqrt{} \)) with a number or expression underneath it. In the context of quadratic equations, radical expressions often appear when solving for the roots (also known as solutions) of the equation. For instance, the square root of a positive number yields a real number, such as \( \sqrt{9} = 3 \).

In contrast, as seen in our exercise, the square root of a negative number, expressed as \( \sqrt{-9} \), is not a real number and thus must be expressed in terms of imaginary numbers. To convey this, we use the radical expression along with the imaginary unit 'i', resulting in solutions like \( 3i \) and \( -3i \) which are the square roots of -9.

Imaginary Numbers

Imaginary numbers, at first glance, might seem like a purely abstract concept, but they are quite useful in various fields of science and engineering. An imaginary number is defined as a multiple of the imaginary unit 'i', which is the square root of -1. Therefore, any time you see \( \sqrt{-1} \), you can replace it with 'i'.

In the context of solving \( m^2 = -9 \), we find that the solution requires taking the square root of a negative number, which results in an imaginary number. The two solutions \( m = 3i \) and \( m = -3i \) both incorporate the imaginary unit 'i', and therefore represent imaginary numbers. By recognizing when to introduce the concept of imaginary numbers, we can solve equations that have no real solutions, extending our mathematical toolkit significantly.