Problem 81 - Algebra 1

Question

Write a question that can be represented by the equation. Then use mental math to solve the equation. $$x^{2}=121$$

Step-by-Step Solution

Answer

There are 11 students in the classroom.

Step 1: Formulate the question

An example of a problem statement could be: 'In a classroom, the square of the number of students is 121. How many students are in the class?' This statement is underlined by the given equation \(x^{2}=121\).

Step 2: Solving the equation using mental math

The equation \(x^{2}=121\) can be solved mentally by finding the number whose square equals 121. This number is 11 as \(11*11 = 121\). It’s worth noting that in the real number system, the equation could also have a negative solution, -11, as \((-11)*(-11) = 121\). The number of students can’t be negative, so in this context only the positive solution is valid.

Key Concepts

Square Root

The square root of a number \(n\) is a value that, when multiplied by itself, gives \(n\). For example, the square root of 121 is the number that satisfies \(x^2 = 121\). To solve this using the square root, we look for \(x\) such that \(x = \pm \sqrt{121}\). Here, \(\sqrt{121} = 11\). The square root function is essential because it helps us find values that satisfy equations involving squares.

Square roots are often used in real-life applications like geometry, physics, and engineering. Calculating square roots mentally or with a calculator can simplify complex problems. Understanding this concept can also enhance mental math skills when dealing with quadratic equations.

Positive Solution

In our exercise, the equation \(x^2 = 121\) illustrates that the square of a number is 121. The positive solution refers to the non-negative value of \(x\) that satisfies this equation. In this case, the positive solution is 11 because \((11 \times 11) = 121\).

Positive solutions are crucial in situations where negative values are not practical or possible, such as counting objects or measuring distance. Here, we consider only the positive root because we are counting students, and it doesn’t make sense to have a negative count. Understanding positive solutions helps ensure accuracy when interpreting real-world situations in mathematical equations.

Negative Solution

Though we often focus on positive solutions, quadratic equations like \(x^2 = 121\) also have negative solutions. These solutions occur because multiplying two negative numbers results in a positive product. Consequently, -11 is also a solution since \((-11) \times (-11) = 121\).

It is important to recognize negative solutions, especially in mathematical contexts where negative numbers are meaningful, such as temperature changes or debts. However, the context is key when deciding which solution to use. In practical applications, knowing that both positive and negative solutions exist enriches mathematical understanding and problem-solving capabilities.