Problem 16 - Elementary Algebra

Question

Factor. $$ 6 x 2-19 x-10 $$

Step-by-Step Solution

Answer

\((6x + 5)(x - 4)\)

Step 1: Identify the Coefficients

Given the quadratic expression \(6x^2 - 19x - 10\), note the coefficients: \(a = 6\), \(b = -19\), and \(c = -10\). This is in the standard form \(ax^2 + bx + c\).

Step 2: Multiply 'a' and 'c'

Multiply \(a\) and \(c\) to get \(6 \times -10 = -60\). We will use this to find factors that sum to \(b\).

Step 3: Find Pair of Factors

Look for two numbers that multiply to \(-60\) and add up to \(-19\). The numbers are \(-24\) and \(5\) because \((-24) \times 5 = -60\) and \((-24) + 5 = -19\).

Step 4: Rewrite the Middle Term

Rewrite \(-19x\) as \(-24x + 5x\), giving a new expression: \(6x^2 - 24x + 5x - 10\).

Step 5: Factor by Grouping

Group the terms: \((6x^2 - 24x) + (5x - 10)\). Factor each group: \(6x(x - 4) + 5(x - 4)\).

Step 6: Factor the Common Binomial

Factor the common binomial \((x - 4)\) from the expression: \((6x + 5)(x - 4)\). This is the factored form of the quadratic expression.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions that include a term with a variable raised to the power of two, usually taking the form \(ax^2 + bx + c\). These expressions play a significant role in mathematics, especially in algebra, because they appear in various contexts such as calculus, physics, and engineering. The standard form of a quadratic expression helps identify the coefficients easier. Here is how the standard form looks:

• \(a\) is the coefficient of \(x^2\).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

Quadratic equations can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Factoring, among these, offers a straightforward route provided the expression is easily factorable.

Factoring by Grouping

Factoring by grouping is a method particularly useful when a quadratic expression is complex, meaning it doesn't easily factor into simple binomials. This method requires the expression to be split into simpler groups that share common factors. Let's break down the steps further:

• Identify two terms from the quadratic that can be grouped together.
• Factor the greatest common factor out of each group separately.
• Check if the remaining terms from each group share a common binomial.

If successful, the expression can be rewritten as a product of binomials, simplifying the process of solving or analyzing quadratic expressions.

Algebraic Coefficients

In quadratic expressions, algebraic coefficients \(a\), \(b\), and \(c\) determine the behavior and solutions of the equation. Recognizing these coefficients is crucial for factoring quadratics.

• \(a\): The coefficient in front of \(x^2\), impacting the parabola's width if graphed.
• \(b\): The coefficient in front of \(x\), affecting the parabola's position related to the y-axis.
• \(c\): The constant term, determining the y-intercept.

Understanding these coefficients aids in implementing strategies like completing the square or using the quadratic formula to solve equations efficiently. Furthermore, knowing their role can assist in predicting algebraic solutions and graph behaviors.

Binomial Factors

Binomial factors are expressions involving the sum or difference of two terms. In the context of factoring quadratics, a quadratic expression might be expressed as a product of such binomials. The factored form helps uncover the roots of the equation, because setting each binomial to zero gives the potential solutions of the quadratic equation. Here's why binomial factors are significant:

• They simplify solving equations.
• They provide insight into the nature of the solutions (real or complex).
• They facilitate understanding of graph behaviors, such as intercepts.

In summary, mastering the use of binomial factors is helpful for efficiently solving quadratic algebraic problems and fully understanding quadratic behaviors.