Problem 73 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ b^{2}-3 b+2=0 $$

Step-by-Step Solution

Answer

Answer: The roots of the given quadratic equation are $$b = \{1, 2\}$$.

Step 1: Identify the quadratic equation

Here, we have a quadratic equation in the variable 'b' given by: $$b^2 - 3b + 2 = 0$$

Step 2: Factor the quadratic equation

To solve this equation, we will try to factor it. Look for two numbers that multiply to 2 (the constant term) and add up to -3 (the coefficient of the linear term). Upon trying different pairs of factors of 2, we find that (-1) and (-2) meet these conditions, since their product is 2 and their sum is -3. Hence, we can factor the given quadratic equation as: $$(b - 1)(b - 2) = 0$$

Step 3: Solve for 'b'

To find the values of 'b', set each factor equal to zero and solve for 'b': 1. \(b - 1 = 0 \Rightarrow b = 1\) 2. \(b - 2 = 0 \Rightarrow b = 2\) So, the solutions for the given quadratic equation are: $$ b = \{1, 2\} $$

Key Concepts

Factoring

Factoring is a method used to simplify expressions and solve equations, especially quadratic equations, by breaking them down into simpler parts. Think of it like finding pieces of a puzzle that fit together to reveal a complete picture. In the context of a quadratic equation such as \(b^2 - 3b + 2 = 0\), factoring means rewriting the equation in the form of \((b - p)(b - q) = 0\), where \(p\) and \(q\) are numbers that meet specific conditions.

To factor the equation, you need two numbers that:

• Multiply to the constant term, which in this case is 2
• Add up to the coefficient of the linear term, here \(-3\)

For this equation, the numbers \(-1\) and \(-2\) meet these conditions. Therefore, we factor it as \((b - 1)(b - 2) = 0\). Factoring is a crucial skill in algebra that makes solving equations much easier.

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true. When dealing with quadratic equations, like \((b - 1)(b - 2) = 0\), the task becomes to find the values of \(b\) that satisfy the equation. Once we factor the quadratic into two binomials, the next step is to apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.

This property lets us set each factor from the equation equal to zero:

• \(b - 1 = 0\)
• \(b - 2 = 0\)

From these, we solve for \(b\):

• \(b = 1\)
• \(b = 2\)

These solutions are the values that make the original equation true. Solving equations by factoring is a powerful technique in algebra that helps simplify complex problems into straightforward solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of equations and inequalities in algebra. In the quadratic equation \(b^2 - 3b + 2 = 0\), the left side is an algebraic expression representing a parabola when graphed.

Each part of the expression plays a crucial role:

• \(b^2\) represents the quadratic term and is responsible for the curve
• \(-3b\) is the linear term and affects the slope and direction
• \(+2\) is the constant term and shifts the graph up or down

Understanding the structure of algebraic expressions helps in operations like factoring, expanding, and simplifying. It also aids in visualizing and interpreting different algebraic problems. Recognizing these components allows for effective manipulation, making algebraic solutions attainable and logical.