Problem 95 - 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: (a) \(528^{2}-527^{2} \quad\) (b) \(122^{2}-120^{2} \quad\) (c) \(1020^{2}-1010^{2}\) Now use the product formula \((A+B)(A-B)=A^{2}-B^{2}\) to evaluate these products in your head: (d) 49\(\cdot 51 \quad\) (e) 998\(\cdot 1002\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Difference of Squares Formula

The difference of squares formula is given by \[A^2 - B^2 = (A + B)(A - B).\] This means a difference between the squares of two numbers can be factored into the product of the sum and the difference of those numbers.

Step 2: Apply the Formula to Problem Part (a)

For part (a), we have \[528^2 - 527^2.\] Let \(A = 528\) and \(B = 527\). Using the formula, \[(528 + 527)(528 - 527) = 1055 \cdot 1 = 1055.\]

Step 3: Solve Problem Part (b)

For part (b), we have \[122^2 - 120^2.\] Let \(A = 122\) and \(B = 120\). Using the formula, \[(122 + 120)(122 - 120) = 242 \cdot 2 = 484.\]

Step 4: Solve Problem Part (c)

For part (c), we have \[1020^2 - 1010^2.\] Let \(A = 1020\) and \(B = 1010\). Using the formula, \[(1020 + 1010)(1020 - 1010) = 2030 \cdot 10 = 20300.\]

Step 5: Use the Product Formula for Problem Part (d)

For part (d), evaluate \(49 \cdot 51\). Recognize that it is difference of squares form: \[(50-1)(50+1) = 50^2 - 1^2 = 2500 - 1 = 2499.\]

Step 6: Use the Product Formula for Problem Part (e)

For part (e), evaluate \(998 \cdot 1002\). Recognize that it is difference of squares form: \[(1000-2)(1000+2) = 1000^2 - 2^2 = 1000000 - 4 = 999996.\]

Key Concepts

Algebra

Algebra is a powerful branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to express general relationships using letters and symbols to represent numbers. This branch of mathematics plays a vital role in solving equations and understanding the properties of geometric shapes. It provides the tools needed to identify patterns, understand changes, and make predictions.

By using algebraic expressions and equations, we can break down complex problems into more manageable parts. Using the difference of squares formula, we can solve algebraic expressions like \( A^2 - B^2 = (A + B)(A - B) \) with ease. This demonstrates algebra's utility in connecting patterns and simplifying calculations.

Factoring

Factoring is the process of breaking down an expression into simpler terms, which when multiplied back together, yield the original expression. Understanding factoring is crucial in algebra because it allows us to simplify expressions and solve equations more efficiently.

The difference of squares is an excellent example of how factoring simplifies expressions. When you have \( A^2 - B^2 \), it can be factored into \((A + B)(A - B)\). This is particularly helpful because it converts a harder multiplication task into an easier addition or subtraction, and multiplication, of smaller numbers. Through this method, we reveal the internal structure of the expression, making it easier to calculate or manipulate further.

Product Formula

The product formula for the difference of squares, \((A+B)(A-B) = A^2 - B^2\), is a handy tool in mathematics, particularly in algebra. It provides a quick way to multiply two binomials by understanding their relationship to the differences formed by their squares.

Importantly, this formula can simplify otherwise complicated multiplications. Consider multiplying 999 by 1001 without a calculator. Recognizing these as \((1000 + 1)(1000 - 1)\), we can easily compute it as \(1000^2 - 1^2\), which simplifies to \(1000000 - 1 = 999999\). This makes seemingly complex problems much simpler, showcasing the incredible utility of algebraic formulas in making calculations swift and less error-prone.

Mental Math

Mental math enables us to perform calculations quickly and efficiently without the aid of calculators or paper. By mastering techniques like the difference of squares and product formulas, one can significantly enhance their ability to calculate mentally.

For instance, when given \( 49 \times 51 \), recognizing it as \((50-1)(50+1)\) helps us understand it's a difference of squares, allowing for a rapid calculation of \( 50^2 - 1^2 = 2499 \). Practicing these techniques can dramatically speed up your math problem-solving skills, strengthening both your analytical abilities and confidence while tackling math challenges.