Problem 41 - Elementary Algebra

Question

Solve using the Square Root Property. \(m^{2}+12=0\)

Step-by-Step Solution

Answer

The solutions are \( m = \/ \pm 2\sqrt{3}i \).

Step 1 - Isolate the squared term

First, subtract 12 from both sides of the equation to isolate the squared term: \[ m^2 + 12 - 12 = 0 - 12 \]which simplifies to\[ m^2 = -12 \]

Step 2 - Apply the Square Root Property

To solve for m, apply the square root to both sides of the equation. Remember to include the positive and negative roots:\[ m = \/ \pm \/ \sqrt{-12} \]Since the square root of a negative number involves the imaginary unit i:\[ m = \/ \pm \/ \sqrt{12}i \]

Step 3 - Simplify the square root

Simplify \( \sqrt{12} \) as follows:\[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3} \]So, the final solution is:\[ m = \/ \pm 2\sqrt{3}i \]

Key Concepts

solving quadratic equations

In mathematics, solving quadratic equations is an essential skill. A quadratic equation generally looks like this: \[ax^2 + bx + c = 0\]. To solve it, one popular method is the Square Root Property, especially when the equation is in the simple form \[ax^2 + c = 0\], with 'b' equal to zero. Here's how it works:

• First, isolate the term containing the square. For example, if your equation is \[m^2 + 12 = 0\], you would subtract 12 from both sides to get \[m^2 = -12\].
• Next, apply the Square Root Property to solve for the variable. This involves taking the square root of both sides, remembering to consider both positive and negative roots: \[m = \/ \pm \sqrt{-12}\].

With these steps, you've used the Square Root Property to find the possible values of 'm' in the equation!

imaginary numbers

Imaginary numbers come into play when taking the square root of a negative number. This can be a confusing concept but is quite fascinating. Here's what you need to know:

• Imaginary numbers are based on the unit 'i', defined as \(i = \sqrt{-1}\).
• For example, the square root of -12 can be expressed as \(\sqrt{-12} = \sqrt{12}i\).
• Imaginary numbers become very useful and even essential in many advanced fields of study such as electrical engineering and physics.

So, when you encounter a square root of a negative number while solving equations, you now know to incorporate 'i' to express it appropriately.

simplifying square roots

Simplifying square roots can make things less complicated and easier to manage. Here's how you do that:

• First, factor the number under the square root into its prime factors. For \( \sqrt{12} = \sqrt{4 \times 3}\).
• Next, find the square roots of the factors. In this case, \( \sqrt{4} = 2\) and \(\sqrt{3}\) cannot be simplified further.
• Combine them to get a simplified form: \( \sqrt{12} = 2\sqrt{3} \).

This method of simplifying helps in getting the solutions in a cleaner and more understandable form, like in our exercise where \(\pm \sqrt{-12}\) simplifies to \(\pm 2\sqrt{3}i\).

complex solutions

Complex solutions come into play when dealing with the square roots of negative numbers. They combine a real part and an imaginary part. Here's how you deal with them:

• A complex number has the form \(a + bi\), where 'a' is the real part and 'bi' is the imaginary part.
• In our exercise, we arrive at complex solutions \(\pm 2\sqrt{3}i\), which can be written as \(0 + 2\sqrt{3}i\) and \(0 - 2\sqrt{3}i\).
• These solutions suggest that there is no real value of 'm' that satisfies the original equation \(m^2 + 12 = 0\).

Understanding complex numbers is important as they frequently appear in various areas of mathematics, engineering, and physics.