Problem 82 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{b^{2}+6 b+9} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: √(b^2 + 6b + 9) Answer: b + 3

Step 1: Factor the Quadratic Expression Inside the Square Root if Possible

To begin, we need to check if we can factor the quadratic expression inside the square root. The expression is: $$ b^2 + 6b + 9 $$ We can rewrite the expression as: $$ (b + 3)^2 $$ Now we have factored the quadratic expression.

Step 2: Simplify the Radical Expression

Now that we have factored the quadratic expression inside the square root, we can simplify the entire radical expression. We have: $$ \sqrt{(b + 3)^2} $$ Since the square root of a square is just the original number, the final simplified expression is: $$ b + 3 $$ So the simplified radical expression is: $$ b + 3 $$

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is a critical skill in algebra that allows for the simplification of equations and easier solving of problems. A quadratic expression is typically written in the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients, and \( x \) denotes the variable. To factor such an expression, we look for two numbers that multiply to give \( ac \) (the product of the first and last coefficients) and add up to \( b \) (the middle coefficient).

In the textbook solution, the quadratic expression \( b^2 + 6b + 9 \) factors neatly into \( (b+3)^2 \), since \( 3 \times \( 3 = 9 \) and \( 3 + 3 = 6 \)—the coefficients of the original expression. Recognizing such patterns can significantly reduce the complexity of solving equations and is an instrumental step in the process of radical simplification.

Radical Simplification

Radical simplification involves the process of finding a simpler or more explicit form of a mathematical expression that contains a root, such as a square root, cube root, etc. The goal is to remove the radical if possible or to minimize the expression underneath the radical.

The square root of a squared quantity, as shown in the exercise \( \sqrt{(b+3)^2} \), simplifies directly to the value inside the square, which is \( b+3 \). That is because the operation of square rooting is the inverse of squaring. However, caution is needed because the square root function actually implies the positive square root. To fully encompass the solution, one would consider both positive and negative possibilities (\( \pm(b+3) \)) if dealing with equations. The simplicity of this step hinges on a well-done factorization; thus, understanding how to accurately factor quadratic expressions is essential for effective radical simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. These expressions become the foundation of most algebraic problems and are manipulated to solve for unknown values. In the given problem, \( b+3 \) is the simplified algebraic expression obtained from factoring and simplifying the original radical expression. While it seems straightforward, it is essential to grasp that each part of an algebraic expression has a role. The variable \( b \) represents an unknown quantity we might solve for, while the number \( 3 \) is a constant that adds a specific value to this unknown.Understanding the relationship between variables and constants is necessary when performing operations like simplification and factorization. Keeping expressions in their simplest form often makes it easier to perform subsequent algebraic operations, whether they are for solving equations, plotting graphs, or applying them to real-world problems.