Problem 67 - Elementary Algebra

Question

Factor completely. $$ 20 x 2-41 x-9 $$

Step-by-Step Solution

Answer

The factored form is \((4x - 9)(5x + 1)\).

Step 1: Identify the Equation

Given quadratic expression is \(20x^2 - 41x - 9\). The goal is to factor it completely.

Step 2: Determine Factorable Format

The expression is in the form \(ax^2 + bx + c\) where \(a = 20\), \(b = -41\), and \(c = -9\). We need to factor it into two binomials.

Step 3: Find Factor Pairs for 'ac'

Calculate the product of \(a\) and \(c\): \(20 \times -9 = -180\). List pairs of factors of \(-180\) that add to \(-41\).

Step 4: Choose the Correct Factor Pair

The correct pair that multiplies to \(-180\) and adds to \(-41\) is \(4\) and \(-45\).

Step 5: Rewrite the Middle Term

Rewrite the expression using \(4\) and \(-45\) for splitting \(-41x\). The expression becomes \(20x^2 + 4x - 45x - 9\).

Step 6: Factor by Grouping

Group the terms: \((20x^2 + 4x) + (-45x - 9)\). Factor each group separately: \(4x(5x + 1) - 9(5x + 1)\).

Step 7: Factor Out the Common Binomial

Factor out \((5x + 1)\) from the entire expression: \((4x - 9)(5x + 1)\).

Step 8: Verify the Factorization

Multiply \((4x - 9)(5x + 1)\): \((4x \cdot 5x) + (4x \cdot 1) + (-9 \cdot 5x) + (-9 \cdot 1) = 20x^2 + 4x - 45x - 9\), which simplifies to the original expression.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions reflecting a polynomial of degree two, typically written in the form \(ax^2 + bx + c = 0\). These equations are called 'quadratic' because 'quad' refers to 'square,' indicating the highest exponent is two.
The constants \(a\), \(b\), and \(c\) represent coefficients, where \(a\) cannot be zero; otherwise, it wouldn't be a quadratic equation.
Quadratic equations can be solved through various methods, including:

• Factoring
• Completing the square
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Factoring is often the preferred method when the quadratic can be easily expressed as a product of two binomials. Here, our primary goal is to rewrite the equation in a factored form through various strategies. A correctly factored quadratic takes the form \((px + q)(rx + s)\).
The roots or solutions to the equation, if factored correctly, are found by setting each binomial to zero, solving for \(x\). Simple solutions make factoring a valuable technique to master, alongside other methods.

Factor by Grouping

Factor by grouping is a method employed to simplify polynomials by reorganizing terms so they can be separated into groups for easy factorization.
This technique proves particularly effective for quadratics and higher order polynomials. When factoring by grouping, follow these steps:

• Identify pairs of terms that can be grouped, each pair having a common factor.
• Extract the greatest common factor (GCF) from each group.
• Identify a common binomial factor between the grouped terms.
• Factor out the common binomial to achieve the simplest expression.

As seen in the example \(20x^2 - 41x - 9\), after finding the suitable factor pairs, replace the middle term \(-41x\) with two terms based on chosen factors. Group these newly formed terms \((20x^2 + 4x) + (-45x - 9)\).
Through factoring by grouping, individual groups are factored to retrieve one common binomial \((5x + 1)\), leading to: \((4x - 9)(5x + 1)\). This approach is efficient, allowing you to see solutions emerge through rearrangement and gradual simplification.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients, involving operations of addition, subtraction, and multiplication, but not division by a variable.
The highest degree of a term in the polynomial establishes its degree. The expression we're factoring, \(20x^2 - 41x - 9\), is a second-degree polynomial.

Common types of polynomials include:

• Monomials - Single term (e.g., \(3x\))
• Binomials - Two terms (e.g., \(3x + 4\))
• Trinomials - Three terms (e.g., \(20x^2 - 41x - 9\))

Factoring polynomials involves expressing them as a product of simpler polynomials. It's a crucial step in solving equations, simplifying expressions, and analyzing roots. For trinomials, looking at the general form \(ax^2 + bx + c\), factorization aids in breaking them into binomials.
Understanding the construction and breakdown of polynomial expressions equips one to handle complex algebraic manipulations with ease. Mastering this concept is foundational for advanced math topics, eventually leading to more profound algebraic theories and analytical techniques.