Problem 96 - Algebra for College Students

Question

Use the pattern \((a-b)^{2}=a^{2}-2 a b+b^{2}\) to compute each of the following numbers mentally, and then check your answers. (a) \(19^{2}\) (b) \(29^{2}\) (c) \(49^{2}\) (d) \(79^{2}\) (e) \(38^{2}\) (f) \(58^{2}\)

Step-by-Step Solution

Answer

(a) 361, (b) 841, (c) 2401, (d) 6241, (e) 1444, (f) 3364.

Step 1: Solve for 19²

We need to rewrite 19 as (20 - 1). Then using the pattern \((a-b)^{2} = a^{2} - 2ab + b^{2} \) with \( a = 20\) and \( b = 1 \):\[ 19^2 = (20 - 1)^2 = 20^2 - 2 \times 20 \times 1 + 1^2 \]Calculate each term:\[ 20^2 = 400 \]\[ -2 \times 20 \times 1 = -40 \]\[ 1^2 = 1 \]Add them together:\[ 400 - 40 + 1 = 361 \]

Step 2: Solve for 29²

29 can be written as (30 - 1). Using the same pattern:Let \( a = 30 \), \( b = 1 \). We have:\[ 29^2 = (30 - 1)^2 = 30^2 - 2 \times 30 \times 1 + 1^2 \]Calculate each term:\[ 30^2 = 900 \]\[ -2 \times 30 \times 1 = -60 \]\[ 1^2 = 1 \]Add them together:\[ 900 - 60 + 1 = 841 \]

Step 3: Solve for 49²

49 can be expressed as (50 - 1). Applying the identity:Let \( a = 50 \), \( b = 1 \). We have:\[ 49^2 = (50 - 1)^2 = 50^2 - 2 \times 50 \times 1 + 1^2 \]Calculate each term:\[ 50^2 = 2500 \]\[ -2 \times 50 \times 1 = -100 \]\[ 1^2 = 1 \]Add them together:\[ 2500 - 100 + 1 = 2401 \]

Step 4: Solve for 79²

Rewrite 79 as (80 - 1) and apply the pattern:Let \( a = 80 \), \( b = 1 \). We have:\[ 79^2 = (80 - 1)^2 = 80^2 - 2 \times 80 \times 1 + 1^2 \]Calculate each term:\[ 80^2 = 6400 \]\[ -2 \times 80 \times 1 = -160 \]\[ 1^2 = 1 \]Add them together:\[ 6400 - 160 + 1 = 6241 \]

Step 5: Solve for 38²

For 38, rewrite it as (40 - 2) and apply the pattern:Let \( a = 40 \), \( b = 2 \). We have:\[ 38^2 = (40 - 2)^2 = 40^2 - 2 \times 40 \times 2 + 2^2 \]Calculate each term:\[ 40^2 = 1600 \]\[ -2 \times 40 \times 2 = -160 \]\[ 2^2 = 4 \]Add them together:\[ 1600 - 160 + 4 = 1444 \]

Step 6: Solve for 58²

Rewrite 58 as (60 - 2) and use the pattern:Let \( a = 60 \), \( b = 2 \). We have:\[ 58^2 = (60 - 2)^2 = 60^2 - 2 \times 60 \times 2 + 2^2 \]Calculate each term:\[ 60^2 = 3600 \]\[ -2 \times 60 \times 2 = -240 \]\[ 2^2 = 4 \]Add them together:\[ 3600 - 240 + 4 = 3364 \]

Key Concepts

Binomial Theorem

The binomial theorem is a powerful mathematical tool that expands expressions raised to a power in the form \((a + b)^n\). However, for the purpose of the exercise given, we focus on a specific case which is \((a-b)^2\). This identity reveals a valuable pattern:

• Start with \( a^2 \), squaring the first term.
• Then, subtract \( 2ab \), which is twice the product of the two terms.
• Finally, add \( b^2 \), which is the square of the second term.

This identity is extremely useful for simplifying calculations like squaring numbers close to a round figure mentally. By rearranging a number as \((a-b)\), and applying this pattern, computation becomes much more straightforward. In the exercise, numbers like 19, 29, 49, etc., near round numbers, are squared using this identity, making the calculations significantly easier.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying mathematical expressions using various algebraic properties and identities. In the context of this exercise, the focus is on manipulating expressions into a form that allows the use of the binomial theorem identity for easy computation. By recognizing that numbers like 19 can be expressed as \(20 - 1\), or 38 as \(40 - 2\), we are effectively simplifying the problem. This rearrangement allows us to apply the identity \((a-b)^2 = a^2 - 2ab + b^2\) effectively. The key is to:

• Break down numbers into the form \((a-b)\), where \(a\) is a round number, and \(b\) is a small integer.
• Use the identity to calculate and combine each term (\(a^2\), \(-2ab\), \(b^2\)) with ease.

Such algebraic manipulation is central to simplifying problems in mathematics and provides a strong foundation for more complex calculations.

Mental Math

Mental math is about performing arithmetic calculations in your head without the assistance of calculators or paper. The exercise is a perfect example of how mental math can be made easier with certain mathematical identities, such as the binomial theorem. Instead of manually computing large numbers like \(19^2\), we break them down using algebraic manipulation into a form that allows quick calculation.By adopting this approach:

• Numbers are rethought in terms of their proximity to round numbers, making calculations manageable.
• The identity \((a-b)^2 = a^2 - 2ab + b^2\) simplifies the arithmetic needed to solve the problem mentally.
• This method can drastically reduce the complexity of larger arithmetic operations, allowing for faster and more efficient computation.

Engaging with mental math in this way not only sharpens your cognitive abilities but also enhances your problem-solving skills, facilitating a stronger grasp on mathematics.