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Magazine Chapter 5: Problem 63
Factor. $$ 16 x^{2}+24 x+5 $$
Short Answer
Expert verified
The factored form is \((4x + 1)(4x + 5)\).
Step by step solution
01
Expand and rewrite the quadratic expression
Identify the quadratic expression to be factored: \[16x^2 + 24x + 5\] This quadratic equation is in the form \(ax^2 + bx + c\) where \(a = 16\), \(b = 24\), and \(c = 5\).
02
Determine two numbers whose product is ac and sum is b
Calculate the product of the coefficient of \(x^2\) term and the constant term: \[16 \times 5 = 80\] Now, find two numbers that multiply to 80 and add to 24.
03
Identify the appropriate pair of numbers
The pairs to consider are: (1, 80), (2, 40), (4, 20), (5, 16), and (8, 10). The pair (4,20) works because: \[4 \times 20 = 80\] and \[4 + 20 = 24\]
04
Rewrite the middle term using the pair
Split the middle term \(24x\) into two terms based on the identified pair (4,20): \[16x^2 + 4x + 20x + 5\]
05
Group and factor by grouping
Group the terms in pairs and factor each pair: \[ (16x^2 + 4x) + (20x + 5) \] Factor out the common terms in each pair: \[ 4x(4x + 1) + 5(4x + 1) \]
06
Factor out the common binomial factor
Both groups contain the common binomial factor \(4x + 1\). Factor out this binomial: \[ (4x + 1)(4x + 5) \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
quadratic equation
A quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \) where \(a\) , \(b\) , and \(c\) are constants, and \(x\) represents an unknown variable. Quadratics are easily identified by the \(x^2\) term, which makes them a second-degree polynomial. These equations can take many forms and can be solved through a variety of methods such as factoring, completing the square, and using the quadratic formula.
factoring
Factoring a quadratic involves rewriting the expression as a product of simpler expressions, called binomials. For example, the quadratic \(16x^2 + 24x + 5\) can be factored into \((4x + 1)(4x + 5)\). This process helps to simplify the equation, making it easier to solve for the values of \(x\). The factoring process includes:
- Rewriting the quadratic equation in the standard \(ax^2 + bx + c\) form.
- Finding two numbers that multiply to the product of \(a\) and \(c\), and add up to \(b\).
- Splitting the middle term based on these numbers and grouping terms.
- Factoring each group and then factoring out the common binomial factor.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication) that represent a particular value. For instance, the expression \(16x^2 + 24x + 5\) is a quadratic expression. Understanding algebraic expressions is fundamental to working with equations because it allows for manipulation and simplification of the mathematical expressions. Key practices include:
- Simplifying terms by combining like terms.
- Using the distributive property to expand or simplify.
- Rearranging terms to better understand their relationships.
binomial factors
Binomial factors are the two-term expressions that are the result of factoring a more complex expression. In the context of quadratic equations, factoring typically results in a product of two binomials. For example, for the expression \(16x^2 + 24x + 5\), the binomial factors are \( (4x + 1) \) and \( (4x + 5) \).
Working with binomial factors involves:
Working with binomial factors involves:
- Identifying common factors in each term group.
- Applying the distributive property in reverse to 'factor out' these common terms.
- Recognizing patterns that can simplify the factoring process, such as special products like the difference of squares.
