Problem 64 - Algebra 1

Question

Completely factor the expression. $$7 x^{2}+8 x+1$$

Step-by-Step Solution

Answer

The factored form of the quadratic expression is \((7x + 1)(x + 1)\).

Step 1: Identify the terms in the quadratic expression

The quadratic expression given is \(7x^2 + 8x + 1\). Here, a = 7, b = 8, and c = 1.

Step 2: Find the pairs

We need to find two numbers e and g such that e*g is equal to c (here, c = 1), and their sum multiplied by a (here, 7) gives b (here, 8). The numbers e, g that satisfy these conditions are 1 and 1 (because 1*1 = 1 and 7*(1 + 1) = 8).

Step 3: Determine d and f

The numbers d and f are both 7 because when d and f are multiplied they give a (which is 7) and it's also stated that d = f since e = g.

Step 4: Write the factors

Using the values of d, e and f, g, the factored form of the quadratic expression is \((7x + 1)(x + 1)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are essential building blocks in algebra. They take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The term with \( x^2 \) is the quadratic term, \( x \) is the linear term, and \( c \) is the constant term. The value of \( a \) determines the parabola's direction—positive \( a \) results in a U-shaped parabola while a negative \( a \) results in an inverted U shape.Understanding the structure of a quadratic expression is crucial when simplifying or factoring it, as it informs how you can break it down into simpler components. This knowledge aids in solving quadratic equations and has applications in various scientific fields.

Factoring Techniques

Factoring is a technique used to simplify quadratic expressions by breaking them down into products of simpler expressions. When factoring \( ax^2 + bx + c \), the goal is to express it in the form \((dx + e)(fx + g)\).Here's how you can factor a quadratic expression like \( 7x^2 + 8x + 1 \):

• Identify two numbers that multiply to \( ac=7*1=7 \) and add up to \( b=8 \).
• Split the linear term using these numbers, then group terms to factor by grouping.
• Rewrite as \((7x + 1)(x + 1)\) shows how each part can be factored.

By practicing different factoring techniques on various quadratic expressions, you'll become more proficient and tackle more complex algebraic problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. They serve as the language for expressing mathematical relationships and principles.Consider the expression \( 7x^2 + 8x + 1 \), incorporating operations like addition and multiplication:

• The term \( 7x^2 \) includes multiplication of 7 and \( x^2 \).
• \( 8x \) involves multiplying 8 by \( x \).
• Addition combines these terms to form the complete expression.

This flexibility allows for a broad range of manipulations, such as simplifying, expanding, or factoring expressions to solve algebraic equations. Understanding how to handle algebraic expressions paves the way for deeper engagement with math and science topics.