Problem 36 - Beginning and Intermediate Algebra

Question

Factor completely. $$q^{2}-49$$

Step-by-Step Solution

Answer

The expression \(q^2 - 49\) factors completely into \((q + 7)(q - 7)\).

Step 1: Identify a and b

In the expression \(q^2 - 49\), a is the base of the first term, which is q, and b is the square root of the second term, which is 7.

Step 2: Apply the difference of squares pattern

Now that we know a and b, apply the difference of squares pattern to get the factored expression: \((a + b)(a - b) = (q + 7)(q - 7)\) The expression \(q^2 - 49\) factors completely into \((q + 7)(q - 7)\).

Key Concepts

Difference of Squares

When factoring polynomials, one useful technique is recognizing patterns like the "difference of squares." This pattern occurs when you have two perfect squares separated by a subtraction sign. It's expressed in the form \(a^2 - b^2\) and can be factored as \((a + b)(a - b)\).

Here's how it works:

• Identify the squares: The first term should be a perfect square, and the second term should also be a perfect square.
• Find their square roots: Determine the square roots of these two terms. These roots are your \(a\) and \(b\) in the formula.
• Apply the formula: Use the pattern to rewrite the expression as \((a + b)(a - b)\).

For example, if you have \(q^2 - 49\), you recognize \(q^2\) as \(q\) squared and 49 as 7 squared. Thus, it becomes \((q + 7)(q - 7)\). This method simplifies the expression by breaking it down into multiplication, making calculations easier.

Quadratic Expressions

Quadratic expressions are a type of polynomial with a degree of two. They are usually written in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

These expressions are significant because they show up in many mathematical contexts. Quadratics can often be factored further, either through:

• Recognizing special patterns, like the difference of squares or perfect square trinomials,
• Using the quadratic formula for more complex cases, or
• Completing the square to find solutions.

In the exercise \(q^2 - 49\), it appears as a simple quadratic expression without the linear \(b\) term. Through recognizing its pattern as a difference of squares, we quickly factor it into two binomials. Understanding quadratics can greatly aid in solving equations and making sense of various algebraic problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols, such as addition or multiplication. They form the basis of all algebraic manipulations and are used to model real-world problems.

These expressions can vary in complexity from simple monomials (like \(3x\)) to complex polynomials (like \(x^3 + 2x^2 - x + 5\)). To manipulate them effectively:

• Understand the rules of arithmetic operations,
• Recognize patterns and identities,
• Factor or simplify expressions when possible, and
• Apply algebraic formulas appropriately.

Factoring is a crucial part of working with algebraic expressions. It breaks down expressions into simpler, more manageable pieces, just as in the example \(q^2 - 49\) is factored to \((q + 7)(q - 7)\). Mastering each component allows students to tackle a wide variety of algebraic problems with confidence.