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The product of a sum and a difference (supplementary material)

Example 1

One pair of opposite sides of a square is increased by \latex{ 3 } \latex{ cm }, while the other pair is decreased by \latex{ 3 } \latex{ cm }, forming a rectangle. Which plane figure has the greater area: the original square or the resulting rectangle? By how much is it greater? (The sides of the square are greater than \latex{ 3 } \latex{ cm }.)

Solution

\latex{ a }

square

rectangle

\latex{ a + 3 }

\latex{ a - 3 }

\latex{ a }

Their sides are: \latex{ a } \latex{ a + 3, \;a-3 }

Their areas are: \latex{ a^2 } \latex{ (a+3)\times (a-3)= \\=a^2+3a-3a-9=a^2-9 }

\latex{ a^2 \gt a^2-9 }

Answer: The area of the original square is \latex{ 9 } \latex{ cm^{2} } greater than the area of the resulting rectangle.

Example 2

Perform the multiplications, combine the like terms, and look for patterns.

a) \latex{ (a-1)\times (a+1) }

b) \latex{ (a-2)\times (a+2) }

c) \latex{ (a-4)\times (a+4) }

Solution

\latex{(a-1)\times (a+1)=a^2+a-a-1=a^2 - 1}
\latex{ (a-2)\times (a+2)=a^2+2a-2a-4= a^2 - 4 }

The middle terms were eliminated.

Guess: \latex{(a-4) \times (a+4) = a^2-4\times{4}=a^2-16}

Check by multiplication:

\latex{ (a-4) \times (a+4) = a^2+4a-4a-16 = a^2 - 16 }

The product of the sum and the difference of two numbers can be written as follows:

\latex{ (a + b) \times (a-b) = a^2-\cancel{ab} + \cancel{ba}-b^2 = a^2- b^2. }

The product of the sum and the difference of two numbers is equal to the difference of their squares, where \latex{a} and \latex{b} are arbitrary rational numbers.

\latex{ (a + b) \times (a-b) = a^2 -b^2 }

Example 3

Write the following products as sums.

\latex{ (n - 1)\times(n + 1) }

\latex{(2 + x) \times (2 - x) }

\latex{(3y + 1) \times (3y - 1)}

\latex{(2b + 3a) \times (3a-2b)}

Solution

\latex{ (n - 1) \times (n + 1) = n^2- 1 }
\latex{ (2 + x) \times (2 - x) = 2^2- x^2 = 4- x^2 }
\latex{ (3y + 1) \times (3y - 1) = (3y)^2-1 = 9y^2- 1}
\latex{ (2b + 3a) \times (3a-2b) = (3a+2b) \times (3a-2b) = (3a)^2- (2b)^2 = 9a^2 - 4b^2 }

Example 4

Without using a calculator, determine

which number is greater: \latex{ 1,000^2 } or \latex{999 \times 1,001,}
the value of \latex{202 \times 198,}
the value of \latex{495 \times 505.}

Solution

Notice that

\latex{ 999 \times 1,001 = (1,000 - 1) \times (1,000 + 1) = 1,000^2 - 1^2 = 1,000^2 - 1 }
Consequently, \latex{ 1,000^2 } is exactly one greater than \latex{ 999 \times 1,001 }.
\latex{202 \times 198=(200+2) \times (200-2)=200^2-2^2=40,000-4=39,996}
\latex{ 495 \times 505 =(500-5) \times (500+5)=500^2-5^2=24,975}

Exercises

{{exercise_number}}. Perform the operations and combine the like terms.

\latex{ (x + 9) \times (x - 9) }

\latex{(1+c)\times(1-c)}

\latex{(4+z)\times(z-4)}

{{exercise_number}}. Write the following products as sums using the identity.

\latex{ (y - 1) \times (y + 1) }

\latex{ (z + 2) \times (z - 2) }

\latex{(2a-1) \times (2a + 1)}

\latex{ \bigg( b + \frac{1}{2} \bigg) \times \bigg( b- \frac{1}{2} \bigg) }

\latex{ (x + y) \times (x - y) }

\latex{ (3 - x) \times (x + 3)}

{{exercise_number}}. Match the corresponding cards and find the one that does not belong.

\latex{ a^2- b^2 }

The square of the sum of two numbers.

The sum of the squares of two numbers.

The difference of the squares of two numbers.

The square of the difference of two numbers.

\latex{ (a - b)(a + b) }

\latex{ (a + b)^2}

\latex{ a^2+ ab + b^2 }

\latex{ a^2+ b^2 }

\latex{ a^2+ 2ab + b^2 }

\latex{ a^2- 2ab + b^2 }

\latex{ (a - b)^2}

{{exercise_number}}. Find a clever way to calculate the following products.

\latex{ 299 \times 301}

\latex{ 305 \times 295 }

\latex{2,002 \times 1,998}

\latex{ 507 \times 493 }

{{exercise_number}}. Answer the questions without calculating the exact values.

The sum of two numbers is \latex{ 128 }, and their difference is \latex{ 48 }. What is the difference of their squares?
The sum of two numbers is \latex{ 69 }, and the difference of their squares is \latex{ 1,449 }. What is their difference?
The difference of two numbers is \latex{ 41 }, and the difference of their squares is \latex{ 2,009 }. What is their sum?

{{exercise_number}}. Solve the following equations.

\latex{(x - 2) \times (x+2) - x^2 = 2x - 10 }

\latex{ x^2+9x = (x - 3)\times (x + 3) }

\latex{(x - 1) \times (x+1) = (x+2)^2 }

\latex{ (x - 2)^2 = (x - 2) \times (x + 2)}

{{exercise_number}}.

Write the product of three consecutive natural numbers in different ways, given that the middle number is \latex{ n }.
Write the product of five consecutive natural numbers in different ways, given that the middle number is \latex{ n }.

{{exercise_number}}. Perform the following calculation. The dots indicate that the terms continue according to the pattern.

\latex{ (100+95) \times (100-95)+(95+90) \times (95-90)+(90+85) \times (90-85)+... }

\latex{ ...+(15+10) \times (15-10)+(10+5) \times (10-5). }

Quiz

Determine the sum of the cubes of the numbers that are not less than \latex{ -9 } and not greater than \latex{ 10 }.

Mathematics 8.

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