Problem 79 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(4 x^{2}+20 x+25\)

Step-by-Step Solution

Answer

The factored form is \((2x + 5)^2\).

Step 1: Understand the Expression

We need to factor the quadratic expression \(4x^2 + 20x + 25\). This means we want to express it as a product of two binomials.

Step 2: Identify the Coefficients

Identify the coefficients of the quadratic equation, where \(a = 4\), \(b = 20\), and \(c = 25\) in the expression \(ax^2 + bx + c\).

Step 3: Check for Perfect Squares

Recognize that both the first term (\(4x^2\)) and the last term (\(25\)) are perfect squares: \((2x)^2\) and \(5^2\), respectively.

Step 4: Verify Middle Term

Verify the relationship of the middle term: Check if \(2ab = b\), where \(a = 2x\) and \(b = 5\). Calculate \(2 \times 2x \times 5 = 20x\), which matches the middle term of the expression.

Step 5: Write Expression as a Square

Since it fits the form \((a+b)^2 = a^2 + 2ab + b^2\), write the expression as a square: \((2x + 5)^2\).

Key Concepts

Quadratic Expressions

Quadratic expressions are mathematical expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is a variable. The highest exponent of the variable is 2, which gives these expressions their name—quadratic.
Quadratic expressions are pivotal in algebra as they form the basis for quadratic equations, which model a variety of real-world phenomena, from projectile motion to business profit analysis.
When factoring quadratic expressions, the goal is to express them as a product of simpler expressions, typically binomials. This process can reveal important characteristics of the expression, such as its roots or solutions, and can simplify calculations in further mathematical applications.

• The standard form of a quadratic expression is \( ax^2 + bx + c \).
• The coefficient \( a \) should not be zero, as this would make it a linear expression.
• Factoring assists in finding solutions to the corresponding quadratic equation.

Understanding quadratic expressions and their factoring is essential for mastering algebraic concepts and solving complex equations.

Perfect Squares

Perfect squares are numbers that are the squares of integers. In the context of algebra, perfect squares often take the form of quadratic expressions. For example, \( 4x^2 \) and \( 25 \) from the exercise are perfect squares: they can be expressed as \((2x)^2\) and \(5^2\), respectively.
Recognizing perfect squares in quadratic expressions helps simplify the factoring process, as it allows the expression to be rewritten as a binomial square. This significantly reduces the complexity of the problem.

• Identify perfect squares in quadratic expressions by looking for terms that can be written as the square of a binomial.
• The expression \( (a + b)^2 \) expands to \( a^2 + 2ab + b^2 \). Notice how the middle term, \(2ab\), helps verify if an expression is a perfect square trinomial.
• Being comfortable with perfect squares aids in recognizing patterns during factoring.

Once you are adept at spotting perfect square terms, rewriting these expressions becomes more intuitive, thus aiding in faster problem-solving.

Binomials

A binomial is an algebraic expression that contains two terms. In the exercise given, we want to factor the quadratic expression into a product of two identical binomials because it forms a perfect square trinomial.
Binomials are useful not only in factoring quadratic expressions but in expanding them as well. Understanding binomials enhances your ability to decompose or construct expressions quickly.

• A typical binomial looks like \( a + b \) or \( a - b \).
• Factoring expressions like \( 4x^2 + 20x + 25 \) involves writing them as \((2x + 5)^2\), which means the binomial \(2x + 5\) is repeated.
• Recognizing common binomials in expressions can simplify complex equations dramatically.

Mastering binomials and their properties is invaluable in tackling various algebraic problems, making them easier to handle and understand.