Problem 72 - Elementary Algebra

Question

Factor using the \(A C\) method. $$ 8 x_{2}+10 x-25 $$

Step-by-Step Solution

Answer

The expression factors to \((4x - 5)(2x + 5)\).

Step 1: Identify Coefficients

The given quadratic expression is \(ax^2 + bx + c\). Here, \(a = 8\), \(b = 10\), and \(c = -25\).

Step 2: Calculate the Product \(ac\)

Multiply \(a\) and \(c\): \(ac = 8 \times (-25) = -200\).

Step 3: Find Two Numbers that Multiply to \(ac\) and Add to \(b\)

We need to find two numbers whose product is \(-200\) and sum is \(10\). These numbers are \(20\) and \(-10\).

Step 4: Rewrite the Middle Term

Rewrite the middle term \(10x\) using the numbers found: \(8x^2 + 20x - 10x - 25\).

Step 5: Factor by Grouping

Group the terms: \((8x^2 + 20x) + (-10x - 25)\). Factor out the greatest common factor from each group: \(4x(2x + 5) - 5(2x + 5)\).

Step 6: Factor out the Common Binomial

Factor out the common binomial \((2x + 5)\): \((4x - 5)(2x + 5)\).

Key Concepts

AC method

The AC method, also known as the "factor by grouping method," is a technique used in factoring quadratic equations. It's especially helpful when the quadratic coefficient, denoted as \(a\), isn't equal to 1. Quadratics follow the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

To use the AC method effectively:

• Multiply \(a\) and \(c\) to find the product, labeled as \(ac\).
• Identify two numbers that multiply to \(ac\) and add up to the coefficient \(b\).
• Break the middle term, \(bx\), using these two numbers.

By rewriting the quadratic in this way, factoring by grouping becomes possible, as we'll see in further sections.

quadratic expressions

Quadratic expressions are polynomial expressions of degree two. Presented in the format \(ax^2 + bx + c\), these expressions are foundational in algebra. They include terms with variables raised to the square, linear, and constant terms.

When dealing with quadratic expressions, the goal is often to factor them completely. Factoring transforms the expression into an equivalent product of binomials or a simpler polynomial form. This process is essential for solving quadratic equations and understanding their roots. By factoring, one can easily find the values of \(x\) that make the equation equal to zero. Finding these solutions will be simplified with techniques like the AC method and factoring by grouping.

factoring by grouping

Factoring by grouping is a crucial step derived from the AC method that allows you to organize and factor expressions easily. After identifying numbers that multiply to the product \(ac\) and sum to \(b\), you split the middle term to start factoring by grouping.

You then form two groups from the quadratic expression:

• The first group consists of terms combined from the original expression. Usually, it includes the term with the highest degree.
• The second group is assembled similarly from the remaining terms

Once grouped, you extract the greatest common factor from each group separately. This step enables you to factor out a common binomial from both groups, allowing a final factor form. For example, in the expression \(8x^2 + 20x - 10x - 25\), splitting it into two groups \((8x^2 + 20x) \) and \((-10x - 25)\) eases the identification of common factors, resulting in the factor form \((4x - 5)(2x + 5)\). This form showcases how factoring by grouping simplifies expressions and helps in solving equations.