Problem 62 - Introductory Algebra for College Students

Question

If 4 times a number is decreased by 3 and this difference is squared, the result is \(9 .\) Find the number \((s)\)

Step-by-Step Solution

Answer

So, the possible values for the unknown number \( s \) are 0 or 1.5

Step 1: Translate the problem statement into an equation

From the problem statement, the equation can be written as \( (4s - 3)^2 = 9 \)

Step 2: Expand the left hand side of the equation

Expand the equation to get \(16s^2 - 24s + 9 = 9\)

Step 3: Simplify the equation

Subtract 9 from both sides to give \(16s^2 - 24s = 0\)

Step 4: Factor the equation for \( s \)

The expression can be factored to give \(16s(s - 1.5) = 0\)

Step 5: Solve for \( s \)

Setting the factors to zero gives two possible solutions: \( s = 0 \) or \( s = 1.5 \)

Key Concepts

Factoring

Factoring is a fundamental skill in algebra that involves breaking down complex expressions into simpler, multiplied terms, or factors. This process makes it easier to solve equations and understand relationships between numbers. The main aim of factoring is to find two or more expressions that multiply together to give the original expression.

Common factoring techniques include:

• Identifying a common factor: Look for a number or variable that can divide each term in the equation.
• Factoring quadratics: Such as using the method of splitting the middle term or using formulas like \((a+b)^2 = a^2 + 2ab + b^2\).
• Difference of squares: Recognize patterns \(a^2 - b^2 = (a+b)(a-b)\).

In our specific problem, after simplifying the original equation, we were able to factor the expression into \(16s(s-1.5) = 0\). This reveals two possibilities where the expression is zero, illustrating how factoring helps in finding solutions.

Algebraic Expressions

Algebraic expressions represent quantities by combining variables, numbers, and mathematical operations. Understanding them is crucial as they are the building blocks of algebra.

An algebraic expression involves:

• Coefficients: Numerical factors of a term, such as 16 in \(16s^2\).
• Variables: Symbols used to represent unknown values, like \(s\).
• Constants: Fixed numbers, like -24, in the equations.
• Operations: Including addition, subtraction, multiplication, and division.

In our quadratic equation \((4s - 3)^2 = 9\), \(4s - 3\) is an algebraic expression that is squared. When expanded, it results in a quadratic form \(16s^2 - 24s + 9\). Recognizing and manipulating these expressions is vital for solving equations.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. This process often requires simplifying the equation and using various techniques to isolate the variable.

Steps typically include:

• Expanding and simplifying expressions to reach a manageable form.
• Rearranging the terms, often by moving them across the equals sign while balancing with inverse operations.
• Factoring, which can simplify solutions for quadratic or higher-power equations.

In the given problem, once the expanded expression \(16s^2 - 24s + 9 = 9\) was simplified to \(16s^2 - 24s = 0\), we solved it by factoring. Setting each factor \(16s = 0\) and \(s - 1.5 = 0\) to zero allowed us to find the potential values of \(s\). These techniques highlight the logical steps required to successfully solve algebraic equations.