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Chapter 9: Problem 33

$$ f(x)=(x+2)^{2}-1 $$

Short Answer

Expert verified
f(x) = x^2 + 4x + 3

Step by step solution

01

- Identify the function

The given function is defined as: \[ f(x) = (x+2)^2 - 1 \]
02

- Expand the quadratic expression

First, expand the term \[ (x+2)^2 \]. Using the formula for the square of a binomial, we have \[ (x+2)^2 = x^2 + 4x + 4 \].
03

- Substitute the expanded form back into the function

Now, replace \[ (x+2)^2 \] with \[ x^2 + 4x + 4 \] in the original function: \[ f(x) = x^2 + 4x + 4 - 1 \].
04

- Simplify the expression

Finally, combine like terms to simplify: \[ f(x) = x^2 + 4x + 3 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Expression
A quadratic expression is a polynomial of degree 2, which means the highest power of the variable is 2. Usually, it has the form
\[ ax^2 + bx + c \], where \(a\), \(b\), and \(c\) are constants. In the given exercise, the function initially looks like
\[ f(x) = (x+2)^2 - 1 \]. To work with it effectively, we need to be comfortable with identifying and manipulating quadratic expressions. The first step is to recognize that
\( (x+2)^2 \) is a binomial squared, which is a classic form of a quadratic expression.
Binomial Expansion
When dealing with quadratic expressions like
\( (x+2)^2 \), we often use binomial expansion to simplify. Binomial expansion involves expanding expressions that are raised to a power, using known algebraic identities. For instance, the formula for the square of a binomial
\( (a+b)^2 = a^2 + 2ab + b^2 \). Applying this to
\( (x+2)^2 \), we set
\( a = x \) and
\( b = 2 \), resulting in the expanded form
\( x^2 + 4x + 4 \). This expansion is crucial because it transforms the binomial into a standard quadratic expression, making it easier to work within further steps.
Function Simplification
Once the quadratic expression is expanded, the next step is function simplification. Simplification involves combining like terms and reducing the expression to its simplest form. After replacing
\( (x+2)^2 \) with
\( x^2 + 4x + 4 \) in the function
\( f(x) = (x+2)^2 - 1 \), we get
\( f(x) = x^2 + 4x + 4 - 1 \). Combining the constants
4 and
-1 gives us
3. So, we obtain the simplified function:
\( f(x) = x^2 + 4x + 3 \). Simplifying functions is crucial because it provides a clearer and more manageable form, which is often needed for further mathematical operations, such as solving equations or graphing.

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