Problem 10 - Elementary and Intermediate Algebra

Question

Factor. \(y^{2}-49\)

Step-by-Step Solution

Answer

The factorization of \(y^2 - 49\) is \((y - 7)(y + 7)\).

Step 1: Identify the expression as a difference of squares

The given expression is \(y^2 - 49\). Notice that \(49\) is a perfect square since it is \(7^2\). The expression is in the form of \(a^2 - b^2\) where \(a = y\) and \(b = 7\).

Step 2: Apply the difference of squares formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Applying this to the expression \(y^2 - 49\), where \(a = y\) and \(b = 7\), we get \((y - 7)(y + 7)\).

Key Concepts

Difference of Squares

The difference of squares is a specific way to factor certain expressions. Generally, this method applies to expressions of the form \(a^2 - b^2\). The term "squares" refers to the fact that both \(a\) and \(b\) are numbers or variables raised to the power of 2—hence, they are "perfect squares."
When we look at \(y^2 - 49\), we can identify it as a difference of squares because \(y^2\) is a square and \(49\) is a square of 7 (since \(7^2 = 49\)). Therefore, our given problem fits this form, with \(a = y\) and \(b = 7\).
The elegant part about this formula, \(a^2 - b^2 = (a - b)(a + b)\), is that it allows us to quickly and convincingly factor these kinds of problems. We separate each square into its components, creating two binomials: one adding \(a\) and \(b\), and one subtracting \(b\) from \(a\). Thus, \(y^2 - 49\) becomes \((y - 7)(y + 7)\).

Perfect Square

Understanding what a perfect square is can immensely aid in recognizing expressions that can be factored using the difference of squares. A perfect square is simply a number or expression that results from squaring an integer or a polynomial. For instance, \(9\) is a perfect square because it equals \(3^2\). Similarly, in algebra, expressions like \(x^2\) or \(9x^2\) are perfect squares.
Knowing this, we can quickly determine that \(49\) is a perfect square, as it equals \(7^2\). Recognizing \(y^2\) in \(y^2 - 49\) also confirms that it's a perfect square. Recognizing the perfect square enables us to apply specific factoring techniques with accuracy, like identifying the difference of squares structure.
In general, understanding perfect squares gives students a handy key: if you can spot them, you often find yourself navigating equations much more easily.

Factoring Quadratic Expressions

Factoring quadratic expressions is a fundamental algebraic skill. It involves re-expressing a quadratic equation (one in the form \(ax^2 + bx + c\)) as a product of its factors. The goal is to break it down into simpler components that, when multiplied together, give the original quadratic expression. This process can simplify solving equations and help in understanding the relationships between different parts of algebraic expressions.
In cases like \(y^2 - 49\), which represents a quadratic expression since it is in the form \(a^2 - c\) (a quadratic with no middle term), recognizing the opportunity to apply the difference of squares pattern is key.
Not all quadratic expressions will fit the difference of squares formula, but many can still be factored to uncover roots or solutions to equations. Always be on the lookout for perfect squares in the expression, as they may be the door to quick and effective factoring using special patterns.