Problem 61 - Introductory Algebra for College Students

Question

If 3 times a number is increased by 2 and this sum is squared, the result is \(49 .\) Find the number(s).

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 5/3\) and \(x = -3\).

Step 1: Formulation of the equation

According to the problem, 3 times a number x is increased by 2 and this sum is squared to give 49. This can be expressed as \((3x + 2)^2 = 49\).

Step 2: Simplification of the equation

Taking square root on both sides, we get the equation in simpler form: \(3x + 2 =\pm \sqrt{49}\) which becomes \(3x + 2 = \pm 7\)

Step 3: Solve the equation for x

Solving the equation \(3x + 2 = 7\), we get \(x = 5/3\). Also, solving the equation \(3x + 2 = -7\), we get \(x = -3\).

Step 4: Verification of the solutions

Substitute \(x = 5/3\) in the original equation. It satisfies the equation. Substitute \(x = -3\) into equation. It also satisfies the equation. Hence, both \( x = 5/3\) and \( x = -3\) are solutions.

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They are combinations of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. In the problem given, we work with an expression like \(3x + 2\), where:

• \(3x\) represents a term involving both a coefficient (3) and a variable (\(x\)), indicating 3 times some unknown number.
• \(+ 2\) is a constant term that modifies the base expression.

To solve problems with such expressions, we often need to perform operations like expanding (for expressions like \((3x + 2)^2\)), simplifying, and factorizing. It's crucial to understand these components to correctly set up and solve equations. Recognizing how terms combine and affect each other visually and arithmetically enhances clarity when solving these expressions.

Solving Equations

Solving equations is the heart of finding unknown values in math. In the provided exercise, solving involves turning \((3x + 2)^2 = 49\) into a form where \(x\) can be isolated:

• First, you simplify the expression by removing powers through operations such as taking square roots.
• Doing so turns the equation into \(3x + 2 = \pm 7\), a more manageable linear formula.

From here, handling the equation means separating out the variable \(x\):

• By solving \(3x + 2 = 7\), you find \(x = \frac{5}{3}\).
• Solving \(3x + 2 = -7\), you discover \(x = -3\).

Verifying each solution by substituting back into the original equation ensures accuracy. This process reinforces understanding, showing each calculated solution as legitimate within the initial context.

Mathematical Problem Solving

Mathematical problem solving is about systematically approaching solutions and understanding the underlying processes. In complex problems like the current exercise, this involves:

• Reading the problem thoroughly to decode what is asked.
• Identifying relevant mathematical operations or transformations – here, realizing when to use squaring or square roots.
• Employing logical reasoning to manipulate expressions and equations appropriately.

A strategic approach includes breaking problems into smaller parts, ensuring each step leads logically to the next. For instance, understanding that \((3x + 2)\) must separately satisfy both of \(7\) and \(-7\) outcomes helps in structuring the sequence of steps. By framing problems through smaller, connected tasks and re-evaluating progress frequently, students can enhance their mathematical thinking and problem-solving skills.