Problem 26 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ m^{2}=22 $$

Step-by-Step Solution

Answer

m = ±√22

Step 1: Understand the Square Root Property

The square root property states that if x^2 = c, then x = ±√c. This property will be used to solve the equation.

Step 2: Apply the Square Root Property

To solve m^2 = 22, take the square root of both sides of the equation: m = ±√22.

Step 3: Simplify the Radicals

Since 22 is not a perfect square and does not have any square factors other than 1, it cannot be simplified further. So, m = ±√22 is the simplified form.

Step 4: Write the Final Solution

The final solution is m = ±√22, which represents two solutions: m = √22 and m = -√22.

Key Concepts

Square Roots

Square roots are an essential concept in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Square roots are often represented using the radical symbol √. It's important to note that every positive number has two square roots: one positive and one negative. For instance, the square roots of 9 are 3 and -3. This dual nature is represented as ±√ in equations. Understanding square roots helps in solving various mathematical problems, especially quadratic equations.

Simplifying Radicals

Simplifying radicals means expressing a radical in its simplest form. A radical is simplified when there are no perfect square factors other than 1 under the radical sign, and no fractions inside the radical. Let's consider the square root of 22. Since 22 is not a perfect square and does not have any square factors other than 1, √22 is already simplified. However, if we had a number like 50, we could simplify it as follows:

• Recognize that 50 = 25 * 2.
• Since the square root of 25 is 5, we can rewrite √50 as √(25*2).
• Simplify this to 5√2.

This process of breaking down a number inside the radical into its prime factors and then simplifying is crucial for many algebraic problems.

Solving Quadratic Equations

One common method for solving quadratic equations is using the square root property. A quadratic equation is typically written in the form ax^2 + bx + c = 0. However, in our exercise, we have an equation in the form m^2 = 22, which is easier to solve using the square root property. Here’s how the method works:

• Start with the equation m^2 = 22.
• Apply the square root property: if x^2 = c, then x = ±√c.
• Take the square root of both sides: m = ±√22.
• Since 22 doesn't simplify further, we keep the radical as it is.

The final solution is m = ±√22, meaning m = √22 and m = -√22. This property helps to quickly solve quadratic equations, especially when they are in a format that can be directly translated into x^2 = c.