Problem 25 - Elementary Algebra

Question

Factor. $$ x 2-20 x+91 $$

Step-by-Step Solution

Answer

The expression factors to \((x - 7)(x - 13)\).

Step 1: Understand the Problem

We need to factor the quadratic expression \(x^2 - 20x + 91\) into two binomials. This means finding numbers that multiply to 91 (the constant term) and add to -20 (the coefficient of the linear term).

Step 2: Determine the Pattern

The standard form of a factored quadratic expression is \((x - a)(x - b)\) where \(a\) and \(b\) are numbers that satisfy the above criteria. We'll look for two numbers whose product is 91 and whose sum is -20.

Step 3: Find Factor Pairs of 91

Find all pairs of factors of 91. They are (1, 91) and (7, 13). Since the product of \(a\) and \(b\) is positive and their sum is negative, both numbers must be negative. Check: \(-7 \times -13 = 91\) and \(-7 + -13 = -20\).

Step 4: Write the Factored Form

Since the factor pair \((-7, -13)\) works, we write the expression as \((x - 7)(x - 13)\).

Step 5: Verify the Solution

Check the factorization by expanding \((x - 7)(x - 13)\) to ensure it equals the original expression. Expanding gives: \(x^2 - 13x - 7x + 91 = x^2 - 20x + 91\), confirming that our factorization is correct.

Key Concepts

Quadratic Expressions

A quadratic expression is a type of polynomial that typically appears in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The expression involves the variable raised to the second power, which qualifies it as 'quadratic'.
Quadratics often describe parabolas when graphed on a coordinate plane. The shape is determined by the equation's parameters:

• The coefficient \(a\) affects the direction and width of the parabola.
• The value of \(b\) influences the location of the vertex.
• The constant term \(c\) shifts the parabola up or down along the y-axis.

In solving quadratics, factoring is a critical technique, which allows us to break down complex expressions into simpler binomial forms. This is essential when analyzing the roots or solutions of the quadratics.

Binomials

Binomials are algebraic expressions containing two terms. In factoring quadratic expressions, our goal is to express the quadratic as a product of two binomials.
The general format for a pair of binomials derived from a quadratic is \((x - a)(x - b)\). Here, \(a\) and \(b\) are specific values that satisfy both the multiplication and addition criteria from the quadratic's constants. This pairing of terms allows us to simplify the expression and more easily understand its properties.
Understanding how to work with binomials offers a gateway to solving equations, finding the intersection of graphs, and understanding factors in advanced algebra.

Expansion Verification

After factoring a quadratic expression into binomials, it is crucial to verify the factorization through expansion. This involves multiplying the binomials to ensure the product matches the original expression.
For instance, given the factorization \((x - 7)(x - 13)\), we check:

• First, expand: \((x - 7)(x - 13) = x^2 - 13x - 7x + 91\)
• Simplify by combining like terms: \(x^2 - 20x + 91\)
• Confirm that the result is identical to the original expression, \(x^2 - 20x + 91\).

This step assures us that our initial factorization was correctly executed. Expansion verification not only checks errors but also enhances understanding of the relationship between products and factors in algebra.