Problem 284 - Elementary Algebra

Question

Factor completely. \(8 x^{2}-9 x-3\)

Step-by-Step Solution

Answer

The factored form is \((2x - 3)(4x + 1)\).

Step 1: Understand the Problem

The objective is to factor the quadratic expression completely. The expression given is \(8x^{2} - 9x - 3\).

Step 2: Identify the Coefficients

Identify the coefficients in the quadratic expression: \(a = 8\), \(b = -9\), and \(c = -3\).The quadratic expression is of the form \(ax^2 + bx + c\).

Step 3: Find Two Numbers that Multiply and Add

We need two numbers that multiply to \(a \times c = 8 \times -3 = -24\) and add to \(b = -9\). These numbers are \(-12\) and \(3\) since \((-12) \times 3 = -24\) and \((-12) + 3 = -9\).

Step 4: Split the Middle Term

Rewrite the quadratic expression by splitting the middle term using the numbers found: \(8x^2 - 12x + 3x - 3\).

Step 5: Factor by Grouping

Group the terms to factor by grouping:\((8x^2 - 12x) + (3x - 3)\). Factor out the greatest common factor from each group: \[4x(2x - 3) + 1(3x - 3)\].

Step 6: Factor Out the Common Binomial Factor

Recognize that \((2x - 3)\) is a common binomial factor: \((2x - 3)(4x + 1)\).

Step 7: Write the Final Factored Form

The completely factored form of the quadratic expression is: \((2x - 3)(4x + 1)\).

Key Concepts

quadratic expressions

Quadratic expressions are a type of polynomial that include terms up to the second degree. The general form is given by: ax^2 + bx + c, where:

• a, b, and c are constants
• x is the variable

In our exercise, we have the quadratic expression 8x^2 - 9x - 3. Here, a = 8, b = -9, and c = -3. To solve such expressions, we often use factoring techniques, which involve rewriting the expression as a product of simpler binomials. This helps in solving equations where the quadratic equals zero, as the binomials provide potential solutions for x.

factoring by grouping

Factoring by grouping involves rewriting and rearranging the quadratic expression into groups that can be factored separately. Here's how we apply this to our quadratic expression step by step:
1. Identify two numbers that multiply to a * c (in our case, 8 * -3 = -24) and add to b (-9). These numbers are -12 and 3.
2. Rewrite the expression by splitting the middle term (bx) using these numbers: 8x^2 - 12x + 3x - 3.
3. Group the terms: (8x^2 - 12x) + (3x - 3).
4. Factor out the greatest common factor (GCF) from each group: 4x(2x - 3) + 1(3x - 3).
Factoring by grouping simplifies complex expressions, making it easier to identify common factors.

algebraic coefficients

Algebraic coefficients are the numerical factors of the terms in an algebraic expression. In our given quadratic expression 8x^2 - 9x - 3, the coefficients are:

• a = 8 (coefficient of x^2)
• b = -9 (coefficient of x)
• c = -3 (constant term)

Understanding the coefficients is crucial for factoring. They determine the values used in finding two numbers that multiply to a * c and add to b. This forms the basis for rewriting and factoring the expression effectively. In our example, knowing the coefficients helped us split the middle term properly and successfully factor the quadratic expression completely.