Problem 101 - Algebra and Trigonometry

Question

The Power of Algebraic Formulas Use the Difference of Squares Formula to factor \(17^{2}-16^{2} .\) Notice that it is easy to calculate the factored form in your head but not so easy to calculate the original form in this way. Evaluate each expression in your head: $$ \begin{array}{l}{\text { (a) } 528^{2}-527^{2} \quad \text { (b) } 122^{2}-120^{2} \quad \text { (c) } 1020^{2}-1010^{2}} \\ {\text { Now use the product formula }(A-B)(A+B)=A^{2}-B^{2} \text { to }} \\ {\text { evaluate these products in your head: }} \\ {\text { (d) } 49 \cdot 51 \quad \text { (e) } 998 \cdot 1002}\end{array} $$

Step-by-Step Solution

Answer

(a) 1055, (b) 484, (c) 20300, (d) 2499, (e) 999996.

Step 1: Recall the Difference of Squares Formula

The Difference of Squares Formula is given by \( (A - B)(A + B) = A^2 - B^2 \). This formula helps in factoring differences of squares quickly.

Step 2: Apply the Formula to 17² - 16²

Recognize that \( 17^2 - 16^2 \) is a difference of squares where \( A = 17 \) and \( B = 16 \). By the formula, we have:\[ (17 - 16)(17 + 16) = 1 \times 33 = 33 \].

Step 3: Simplify 528² - 527² Using the Formula

For \( 528^2 - 527^2 \), set \( A = 528 \) and \( B = 527 \). Thus, \[(528 - 527)(528 + 527) = 1 \times 1055 = 1055 \].

Step 4: Simplify 122² - 120² Using the Formula

Using \( A = 122 \) and \( B = 120 \), we get: \[ (122 - 120)(122 + 120) = 2 \times 242 = 484 \].

Step 5: Simplify 1020² - 1010² Using the Formula

With \( A = 1020 \) and \( B = 1010 \), apply the formula: \[ (1020 - 1010)(1020 + 1010) = 10 \times 2030 = 20300 \].

Step 6: Evaluate 49 x 51 Using the Product Formula

Set \( A = 50 \) and calculate: \( (50-1)(50+1) = 50^2 - 1^2 = 2500 - 1 = 2499 \).

Step 7: Evaluate 998 x 1002 Using the Product Formula

Let \( A = 1000 \), then \( (1000-2)(1000+2) = 1000^2 - 2^2 = 1000000 - 4 = 999996 \).

Key Concepts

Difference of Squares

The Difference of Squares is an important algebraic formula. It is particularly useful for simplifying expressions. The formula is represented as \((A - B)(A + B) = A^2 - B^2\). This expression indicates that the product of the sum and difference of two numbers equals the difference of their squares.

This formula can be applied when you encounter a subtraction of two squared terms. Recognizing this pattern allows you to factor the expression efficiently. For example, in the expression \(17^2 - 16^2\), setting \(A = 17\) and \(B = 16\) enables you to rewrite it as \((17 - 16)(17 + 16)\), which simplifies to \(1 \times 33 = 33\). The difference of squares is a powerful tool when dealing with quadratic expressions.

Factoring

Factoring involves breaking down a complex expression into simpler components, or factors, that when multiplied together give the original expression. This is a key concept in algebra that often involves recognizing patterns, like the difference of squares.

In the given exercise, factoring using the difference of squares helped simplify expressions like \(528^2 - 527^2\). By identifying \(A = 528\) and \(B = 527\), we reduced it to \((528 - 527)(528 + 527)\), which transforms a potentially tough calculation into a simple product of \(1 \times 1055 = 1055\). Factoring can thus make solving algebraic problems more straightforward by reducing the work needed to reach a solution.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In its simplest form, algebra involves solving equations and simplifying expressions. By using symbols like \(A\) and \(B\), we can generalize mathematical operations.

Algebra can seem complex at first, but with practice, it allows for efficient problem-solving. For instance, in the exercises, recognizing patterns such as the difference of squares allows for the simplification of complex expressions into simpler calculations. These concepts form a foundation on which more advanced algebraic techniques are built, providing a toolkit for tackling a variety of mathematical problems.

Product Formulas

Product formulas, such as the Difference of Squares formula, are equations that express the product of terms in a simplified form. They are crucial when calculating large products mentally, as they reduce the computational load by transforming a multiplication task into a simpler subtraction.

For example, expressions like \(49 \times 51\) become much easier to evaluate by setting \(A = 50\) and using the fact that \((A - 1)(A + 1) = A^2 - 1^2\). This simplifies \(49 \times 51\) to \(2500 - 1 = 2499\). Using product formulas can make mental calculations quicker and less error-prone, especially when dealing with numbers that are close together or involve squaring.