Problem 46 - 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. $$ m^{2}-12=52 $$

Step-by-Step Solution

Answer

The solutions to the original equation are \( m = 8 \) and \( m = -8 \)

Step 1: Rearrange the Equation

To isolate \( m^{2} \), add 12 to both sides of the equation. This results in the equation \( m^{2}=52+12 \) or \( m^{2}=64 \)

Step 2: Take Square Root

Next, to solve for \( m \), take the square root of both sides of the equation. This results in the equation \( m=\sqrt{64} \).

Step 3: Simplification

Simplify the right side of the equation to find \( m \). Since the square root of 64 is 8, the equation \( m = \pm 8 \) meaning \( m \) equals either 8 or -8.

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and encapsulate numbers, variables, and arithmetic operations. A fundamental concept to understand is the meaning of variables, which are symbols (often letters) standing in for unknown values. In the provided exercise, the variable is represented by the symbol m, and the expression is m^2 - 12 = 52. This is a quadratic equation, which is a type of algebraic expression that includes a variable raised to the second power.

When solving quadratic equations, the goal is to isolate the variable and determine its value(s). The first step in the solution demonstrated the initial manipulation of the expression, involving the addition of 12 to both sides to isolate m^2, a crucial step towards finding the solution.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, in the exercise, we are dealing with the square root of 64. One could ask, 'What number, when multiplied by itself, equals 64?'. The answer is 8 since 8 x 8 = 64. The square root operation is the inverse of squaring a number.

In the context of quadratic equations, taking the square root is a typical method to solve for the variable once the equation is in the form of m^2 = some number. However, it's important to note that every positive number actually has two square roots: a positive and a negative root. This is why the solution to the exercise includes both m = +8 and m = -8, as both numbers squared will result in 64.

Radical Expressions

Radical expressions involve roots of numbers, including square roots, cube roots, and so on. The symbol for the square root, \(\sqrt{ }\), is also known as the radical symbol, hence the name 'radical expressions'. In cases where square roots lead to non-integer solutions, it is necessary to express the answer in its radical form.

For instance, if the equation were m^2 = 50, the square root of 50 cannot be simplified to an integer. Thus, the solution would remain in radical form: m = \(\pm\sqrt{50}\). Radical expressions should be simplified as much as possible, as exemplified by the solution to the original exercise. However, in cases where there are no perfect square factors, the expression is left in its radical form. It's a critical skill in algebra to manage and manipulate these expressions, whether to simplify them or to translate between radical and exponential forms.