Problem 52 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$m^{2}=81$$

Step-by-Step Solution

Answer

The solutions are \( m = 9 \) and \( m = -9 \).

Step 1: Understand the Equation

The given equation is \( m^2 = 81 \). This equation indicates that we need to find the value(s) of \( m \) which, when squared, equals 81.

Step 2: Take the Square Root

To solve for \( m \), take the square root of both sides of the equation. This gives \( m = \pm \sqrt{81} \).

Step 3: Solve the Square Root

Calculate \( \sqrt{81} \). Since 81 is a perfect square, \( \sqrt{81} = 9 \). Therefore, \( m = \pm 9 \).

Step 4: Interpret the Solution

The solutions to the equation \( m^2 = 81 \) are \( m = 9 \) and \( m = -9 \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics that help solve equations involving squares of numbers. Essentially, if you have a number, like 81, and you want to find its square root, you're looking for another number that, when multiplied by itself, results in 81. In more mathematical terms, the square root of a number \( x \) is a number \( y \) such that \( y^2 = x \).
When taking the square root of both sides of an equation like \( m^2 = 81 \), it's important to realize it helps you simplify the equation, allowing you to isolate \( m \). The square root process is vital in extracting possible values of \( m \) from the quadratic equation. A square root has both positive and negative versions, leading us to our next topic.

Perfect Squares

A perfect square is an integer that is the square of another integer. For example, 81 is a perfect square because it equals \( 9^2 \). Recognizing perfect squares can significantly simplify solving quadratic equations. In mathematical terms, if a number is a perfect square, it can be expressed as \( n^2 \) for some integer \( n \).When you solve an equation like \( m^2 = 81 \), knowing 81 is a perfect square allows you to immediately state that \( m \) could be \( 9 \) or \(-9\). Otherwise, you'd need to perform a longer calculation process. Identifying perfect squares helps streamline solving these types of equations significantly.

Positive and Negative Solutions

When solving quadratic equations, it is crucial to remember that both positive and negative values can satisfy the equation. In our original exercise solving \( m^2 = 81 \), upon taking the square root, you must consider both \( +9 \) and \( -9 \) as solutions. This is because both \( 9^2 \) and \((-9)^2\) equal 81.
The reasoning is grounded in the property of squares: multiplying two negative numbers gives a positive result as well as multiplying two positive numbers. Therefore, when solving any squared equation, always acknowledge the existence of dual solutions unless specifically restricted by the problem conditions.

Mathematical Reasoning

Mathematical reasoning is the logical thought process used to solve mathematical problems and to justify solutions. When we look at the problem \( m^2 = 81 \), the mathematical reasoning involves understanding the implications of the equation, such as recognizing the need to take square roots and consider both positive and negative outcomes.Effective mathematical reasoning includes:

• Breaking down the problem: Identifying what the equation represents.
• Logical progression: Applying the correct operations, like taking the square root.
• Interpretation: Distinguishing between the possible solutions and their meanings.

It helps ensure the solution is both accurate and comprehensive, reinforcing the correct application of concepts like square roots and recognizing perfect squares. Strong reasoning skills are essential for solving not only specific problems but also for tackling a wide array of mathematical challenges.