Problem 25 - Elementary and Intermediate Algebra

Question

Factor. See Example 1 or Example 6. $$ 7 x^{2}+18 x+11 $$

Step-by-Step Solution

Answer

The expression factors to \((7x + 11)(x + 1)\).

Step 1: Identify coefficients

The given quadratic expression is \( 7x^2 + 18x + 11 \). Here, \( a = 7 \), \( b = 18 \), and \( c = 11 \).

Step 2: Multiply 'a' and 'c'

Multiply the coefficient of \( x^2 \) (\( a \)) and the constant term (\( c \)). \( 7 \times 11 = 77 \).

Step 3: Find factor pair of 77

Look for two numbers that multiply to 77 and add to 18. The numbers are 7 and 11 because \( 7 \times 11 = 77 \) and \( 7 + 11 = 18 \).

Step 4: Rewrite the middle term

Rewrite the original quadratic expression by breaking the middle term into two terms using the numbers found in Step 3. The expression becomes \( 7x^2 + 7x + 11x + 11 \).

Step 5: Factor by grouping

Group the terms: \( (7x^2 + 7x) + (11x + 11) \). Factor each group separately. From the first group, factor out \( 7x \) to get \( 7x(x + 1) \). From the second group, factor out \( 11 \) to get \( 11(x + 1) \).

Step 6: Combine grouped terms

After factoring, the expression becomes \( 7x(x + 1) + 11(x + 1) \). Notice that \( (x + 1) \) is common in both terms. Thus, factor out \( (x + 1) \) to get \( (7x + 11)(x + 1) \).

Key Concepts

Coefficients in Quadratics

In a quadratic expression like \( ax^2 + bx + c \), the letters \( a \), \( b \), and \( c \) are known as coefficients. These coefficients are crucial because they determine the shape and position of the parabola represented by the quadratic expression when graphed. The coefficient \( a \) is called the leading coefficient as it multiplies the squared term. Coefficients \( b \) and \( c \) affect the linear and constant parts of the expression, respectively.
Understanding coefficients helps in various methods of factoring quadratics. For instance, identifying them is the first step in factoring by grouping, as it involves rearranging and breaking down the quadratic expression based on these numbers. In our example, \( a = 7 \), \( b = 18 \), and \( c = 11 \), which clearly outline the expression \( 7x^2 + 18x + 11 \).

• \( a \) influences the width and direction of the parabola.
• \( b \) impacts the slope and position of the vertex.
• \( c \) determines the y-intercept.

By knowing these coefficients, you can proceed with specific factoring strategies, such as factoring by grouping.

Factoring by Grouping

Factoring by grouping is a reliable method for solving quadratic expressions, especially when the expression doesn't factor easily through simple observation or when there's no obvious greatest common factor. This technique involves splitting a quadratic expression into two pairs, both containing a common factor, and then extracting this factor to simplify further. Let's explore how this was applied in the example:
Initially, you multiply the coefficients \( a \) and \( c \), in this case, \( 7 \times 11 = 77 \). You then look for two numbers that multiply to give 77 and add up to 18, which are 7 and 11. These numbers allow the middle term (18x) to be split into two terms: 7x and 11x.
The expression \( 7x^2 + 18x + 11 \) becomes \( 7x^2 + 7x + 11x + 11 \) by rewriting the middle term. Next, you separate these into two groups: \( (7x^2 + 7x) \) and \( (11x + 11) \).

• From the first group, factor out \( 7x \) to get \( 7x(x + 1) \).
• From the second group, factor out 11 to get \( 11(x + 1) \).

Now you have \( 7x(x + 1) + 11(x + 1) \). A common factor \( (x + 1) \) emerges, allowing the quadratic to be simplified further to \( (7x + 11)(x + 1) \). This efficient technique helps when direct factoring is challenging or when dealing with more complex quadratics.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients and \( x \) is the variable. These expressions describe parabolic curves in the Cartesian plane, with the squared term \( x^2 \) indicating their fundamental characteristic—a curve that opens upward or downward depending on the direction of \( a \).
In our example, the expression \( 7x^2 + 18x + 11 \) falls under this category. Factoring quadratic expressions is often vital for solving equations, finding roots, and simplifying expressions for further mathematical manipulations.
Understanding the format and behavior of quadratics helps in graphing and solving them. When graphed, quadratics form a "U" or inverted "U" shape. This visualization can aid in interpreting real-world situations modeled by these expressions, such as projectile motion or area problems.
By mastering techniques like factoring by grouping, students equip themselves with powerful tools for decomposing complex expressions into simpler binomials. This not only aids in solving equations but also opens the door to deeper mathematical understanding and application across various fields. In essence, learning to work with quadratic expressions cultivates a foundational math skill useful in both academic and practical scenarios.