Problem 23 - Elementary and Intermediate Algebra

Question

Factor. See Example 3 or Example \(10 .\) $$x^{2}+5 x-24$$

Step-by-Step Solution

Answer

The factored form of \(x^2 + 5x - 24\) is \((x - 3)(x + 8)\).

Step 1: Identify the Coefficients

For the quadratic equation \(x^2 + 5x - 24\), identify the coefficients: \(a = 1\), \(b = 5\), \(c = -24\). This will help us in the next steps where we need to factor the equation.

Step 2: Set up the Product and Sum

We need two numbers that multiply to \(c = -24\) (product) and add up to \(b = 5\) (sum).

Step 3: Find the Pair of Numbers

After considering pairs of factors of \(-24\), we find that \(8\) and \(-3\) satisfy the conditions: \(8 \times (-3) = -24\) and \(8 + (-3) = 5\).

Step 4: Rewrite the Middle Term

Rewrite the middle term \(5x\) using the numbers found: \(x^2 + 8x - 3x - 24\).

Step 5: Factor by Grouping

Group the terms in pairs and factor each group: \((x^2 + 8x) + (-3x - 24)\).

Step 6: Factor Each Group

Factor out the greatest common factor from each group: \(x(x + 8) - 3(x + 8)\).

Step 7: Combine as a Product of Binomials

Notice \((x + 8)\) is common in both terms, so factor it out: \((x - 3)(x + 8)\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial that involves terms up to the second degree, meaning the highest exponent of the variable is 2. In its standard form, it appears as \( ax^2 + bx + c = 0 \), where "\(a\)", "\(b\)", and "\(c\)" are constants known as coefficients, and "\(x\)" is the variable. Quadratic equations are prevalent in many fields, including physics, engineering, and economics because they model processes with accelerated change.
Understanding the structure of a quadratic equation is crucial because it lays the groundwork for solving or factoring it.

• "\(a\)" is the leading coefficient and cannot be zero because the presence of \(x^2\) is what makes the equation quadratic.
• "\(b\)" is the linear coefficient, which influences the slope of the parabola the equation forms when graphed.
• "\(c\)" is the constant term, which shifts the graph up or down on a coordinate plane.

Recognizing when an equation fits this format enables various solving methods such as the quadratic formula, completing the square, or factoring. In this exercise, you'll use factoring, a method often favored for its simplicity when the equation coefficients are easy to manage.

Coefficients

Coefficients are constant numbers that multiply variables in an algebraic expression or equation. In the quadratic equation \( ax^2 + bx + c \), the coefficients \(a\), \(b\), and \(c\) play key roles.

In the exercise \(x^2 + 5x - 24\), we have the coefficients:

• \(a = 1\), which corresponds to \(x^2\).
• \(b = 5\), the coefficient next to \(x\).
• \(c = -24\), the constant term that doesn't have a variable.

Each of these coefficients affects the shape and position of the quadratic curve if you were to graph it. The value of \(a\) controls the parabola's opening direction and width—the larger it is, the steeper the parabola. \(b\) impacts where the peak or valley of the graph occurs along the x-axis, and \(c\) affects where it crosses the y-axis. While dealing with quadratic factorization, especially in the method of factoring by grouping, identifying these coefficients is the first step, as it helps determine the factors of the quadratic expression.

Factoring by Grouping

Factoring by grouping is a technique used to simplify polynomials by separating terms into groups that can be factored individually, leading to a simpler expression. It's especially useful in situations where other methods of factoring, like using the greatest common divisor, are not feasible.
Here's how it works with our exercise: First, rewrite the quadratic equation by splitting the middle term using two numbers that multiply to give the product of "\(a\) and "\(c\) but add up to "\(b\). In \(x^2 + 5x - 24\), we looked for two numbers that multiply to -24 and add up to 5. Those numbers are 8 and -3. Thus, we rewrite the middle term as \(8x\) and \(-3x\). The expression becomes:\[ x^2 + 8x - 3x - 24 \]

Next, group the terms: \((x^2 + 8x)\) and \((-3x - 24)\). Factor each group individually:

• For \(x^2 + 8x\), factor out \(x\): \(x(x + 8)\).
• For \(-3x - 24\), factor out \(-3\): \(-3(x + 8)\).

Notice \((x + 8)\) is common in both groups, enabling us to factor it out as a common binomial: \((x + 8)(x - 3)\).
This step concludes the factorization using grouping, transforming the complex quadratic into a product of simpler binomials. Factoring by grouping may take practice to master, but it's a powerful tool for tackling a variety of polynomial equations efficiently.