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The square of a sum or a difference (supplementary material)
Certain products follow specific patterns when expressed as sums. It is worth memorising and applying these patterns to calculate more quickly.
Example 1
Try the following magic trick.
- Think of a number.
- Add \latex{5} to it.
- Square the result.
- Subtract ten times the number you thought of.
- Add \latex{5}.
- Subtract the square of the number you first thought of.
I can tell what you got.
How did I know?
Solution
Let the number you thought of be denoted by \latex{ x } and follow the instructions step by step.
- \latex{x}
- \latex{x+5}
- \latex{(x+5)^2}
- \latex{(x+5)^2 - 10x}
- \latex{(x+5)^2 - 10x+5}
- \latex{(x + 5)^2 - 10x + 5 - x^2 = (x + 5)(x + 5) - 10x + 5 - x^2 =} \latex{ = x^2+ 5x + 5x + 25 - 10x+5- x^2 = 10x - 10x + 25 + 5 = 30}
Answer: The result is always \latex{ 30 }, regardless of the number you chose.
Example 2
Write the following algebraic expressions as sums and look for patterns.
a) \latex{ (a+1)^2 }
b) \latex{ (a+2)^2 }
c) \latex{ (a+3)^2 }
d) \latex{ (a+4)^2 }
e) \latex{ (a+6)^2 }
Solution
- \latex{ (a+1)^2= (a+1) \times (a+1) = a^2+a+a+1 = a^2+2a + 1}
- \latex{ (a+2)^2= (a+2) \times (a+2) = a^2+2a+2a + 4 = a^2+4a + 4}
- \latex{ (a+3)^2= (a+3) \times (a+3) = a^2+3a+3a + 9 = a^2+6a + 9}
- \latex{ (a+4)^2= (a+4) \times (a+4) = a^2+4a+4a+16 = a^2+8a + 16}
- Guess: \latex{ (a+6)^2= a^2+2 \times 6a+6^2 = a^2+12a + 36}
Check by expanding the brackets.
\latex{ (a+6)^2= (a+6) \times (a+6)=a^2+6a+6a+6 \times 6=a^2+2 \times 6a+6^2}
The square of the sum of two terms:
\latex{ (a + b)^2= (a + b) \times (a + b) = a^2+ ab + ba + b^2= a^2+2ab + b^2. }
The square of the sum of two terms can be illustrated by a square with sides of length \latex{ a + b }. Its area can be expressed in two different ways:
\latex{b^2}
\latex{a + b}
\latex{a + b}
\latex{(a + b)^2}
\latex{a}
\latex{a}
\latex{b}
\latex{b}
\latex{ab}
\latex{ab}
\latex{a^2}
\latex{ A = (a + b)^2 } \latex{ A = a^2 + ab + ab + b^2 = a^{2}+2ab + b^2 }
The two areas are equal:
\latex{ (a + b)^2= a^2+2ab + b^2. }
The square of the sum of two terms is equal to the sum of the square of the first term, twice the product of the two terms, and the square of the second term (where \latex{a} and \latex{b} are arbitrary rational numbers).
\latex{ (a + b)^2 = a^2+2ab + b^2 }
An identity refers to an equation that is always true, no matter what values from the domain are substituted for the variables.
Examples of identities:
\latex{ 2x + 3x = 5x,\;\;\;\; (a + b)^2= a^2+2ab + b^2,\;\;\;\;a(b + c) = ab + ac. }
Not an identity:
\latex{ 3x = 6 }, since it is only true for the value \latex{ x = 2 }.
(the domain is the set of rational numbers)
Example 3
Square the following algebraic expressions using the identity.
- \latex{ (a + 10)^2 }
- \latex{ (2b+1)^2 }
- \latex{ \bigg( c+\frac{1}{2} \bigg)^2 }
- \latex{ (4d+3e)^2 }
Solution
- \latex{ (a + 10)^2= a^2+2 \times a \times 10 + 10^2 = a^2+20a + 100}
- \latex{ (2b+1)^2= (2b)^2+2\times 2b \times1 + 1^2= 4b^2+4b+1}
- \latex{ \bigg( c+\frac{1}{2} \bigg)^2 = c^2 +2 \times c \times \frac{1}{2} + \bigg( \frac{1}{2} \bigg)^2 = c^2 + c + \frac{1}{4}}
- \latex{ (4d+3e)^2= (4d)^2+2\times 4d \times 3e + (3e)^2= 16d^2+24de+9e^2 }
The square of the difference of two terms can be written in a similar way.
\latex{ (a-b)^2= (a-b) \times (a-b) = a^2-ab - ba + b^2= a^2- 2ab + b^2. }
The square of the difference of two terms is obtained by subtracting twice the product of the two terms from the square of the first term, and then adding the square of the second term (where \latex{a} and \latex{b} are arbitrary rational numbers).
\latex{ (a - b)^2 = a^2 - 2ab + b^2 }
The identity can be illustrated with a square. Observe how the area of the square changes when each side is reduced by \latex{\large b }. If you cut a rectangle of area \latex{\large ab } from the square, then add a square of area \latex{\large b^{2} }, and finally cut another rectangle of area \latex{\large ab}, the resulting square will have an area of \latex{\large (a-b)^{2} }.
\latex{a-b}
\latex{a}
\latex{a}
\latex{a-b}
\latex{b}
\latex{b}
\latex{b}
\latex{a}
\latex{b}
\latex{b}
\latex{a-b}
\latex{a-b}
\latex{ a^2\,\,\,\,\,\,\,\,\,\,\,\,\,\, -\,\,\,\,\,\,\,\,\, ab\,\,\,\,\,\,\,\,\,\, +\,\,\,\,\,\,\,\,\,\,\,\,\, b^2 \,\,\,\,\,\,\,\,\,\,\,\,- ab\,\,\,\,\,\,\,\,\,\,\,\, =\,\,\,\,\,\,\,\,\,\,\,\,\,\, (a-b)^2 }
Example 4
Calculate the following squares as quickly as possible without using a calculator.
a) \latex{ 101^{2} }
b) \latex{ 29^{2} }
c) \latex{ 199^{2} }
Solution
- \latex{ 101^2= (100 + 1)^2 = 100^2+2 \times 100 \times 1 + 1^2 = 10,000 + 200 + 1 = 10,201 }
- \latex{ 29^2= (30 - 1)^2 = 30^2- 2 \times 30 \times 1 + 1^2 = 900 - 60 + 1 = 841 }
- \latex{ 199^2= (200 - 1)^2 = 200^2- 2 \times 200 \times 1 + 1^2 = 40,000 - 400 + 1 = 39,601 }
Example 5
The product of two numbers is \latex{ 56 }, and the sum of their squares is \latex{ 113 }. What is the square of their sum?
Solution
The two numbers are \latex{ a } and \latex{ b }.
Their product is \latex{ a \times b = 56. }
The sum of their squares is \latex{ a^2+ b^2= 113. }
Question: the square of their sum, \latex{ (a + b)^2. }
Their product is \latex{ a \times b = 56. }
The sum of their squares is \latex{ a^2+ b^2= 113. }
Question: the square of their sum, \latex{ (a + b)^2. }
According to the identity for the square of the sum of two numbers:
\latex{ (a + b)^2= a^2+2ab + b^2= 2ab + (a^2+ b^2) = 2 \times 56 + 113 = 225. }
Answer: The square of the sum of the two numbers is \latex{ 225 }.
Some questions can be answered by rearranging equations, even without knowing the exact values of the variables.
Example 6
Are there two consecutive natural numbers whose squares give a differ by a) \latex{ 87 }, b) \latex{ 90? }
Solution
Denote the smaller number by \latex{ n } and its square by \latex{ n^{2} }.
The consecutive natural number after \latex{ n } is \latex{ n+1 }; its square is \latex{ (n+1)^{2} }.
The difference of their square is \latex{ (n+1)^2- n^2 = n^2+ 2n+1- n^2= 2n+1. }
The consecutive natural number after \latex{ n } is \latex{ n+1 }; its square is \latex{ (n+1)^{2} }.
The difference of their square is \latex{ (n+1)^2- n^2 = n^2+ 2n+1- n^2= 2n+1. }
a) The difference of the two consecutive square numbers is \latex{ 87 }.
\latex{ 2n+1 = 87 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;/ -1 }
\latex{ 2n = 86 \;\;\;\;\;\;\;\;\;\;/ \div 2 }
\latex{\underline{\underline{n = 43}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}
\latex{ 2n = 86 \;\;\;\;\;\;\;\;\;\;/ \div 2 }
\latex{\underline{\underline{n = 43}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}
Check: \latex{ 44^{2} - 43^{2} = 1,936 - 1,849 = 87. }
Answer: The two natural numbers that meet the requirements are \latex{ 43 } and \latex{ 44 }.
b) The difference of the two consecutive squares is \latex{ 90 }.
\latex{ 2n+1 = 90 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;/ -1 }
\latex{ 2n = 89 \;\;\;\;\;\;\;\;\;\;/\div 2 }
\latex{\underline{\underline{n = 44.5}},\;\;\;\;\;\;\;\;\;\;\;\;}
\latex{ 2n = 89 \;\;\;\;\;\;\;\;\;\;/\div 2 }
\latex{\underline{\underline{n = 44.5}},\;\;\;\;\;\;\;\;\;\;\;\;}
which is not a natural number.
Answer: There are no two consecutive natural numbers whose squares differ by \latex{ 90. }
\latex{n^2}
\latex{1}
\latex{1}
\latex{1}
\latex{n}
\latex{n}
\latex{n}
\latex{n}
\latex{ (n+1)^2-n^2=2n+1 }
The difference of the squares of two consecutive natural numbers cannot be even, since the algebraic expression \latex{ (n+1)^2 - n^2= 2n + 1 } always gives an odd result when \latex{ n } is a natural number.
Exercises
{{exercise_number}}. Square the following algebraic expressions.
- \latex{ (a + 10)^2 }
- \latex{ (b+8)^2 }
- \latex{(x + y)^2}
- \latex{ (1 + 2x)^2 }
- \latex{ \bigg( 2y+\frac{1}{2} \bigg)^2 }
- \latex{ (a+2b)^2}
- \latex{ (3x+2y)^2}
- \latex{ (10a+10b)^2}
{{exercise_number}}. Write algebraic expressions to express the areas of the following squares in different ways.
\latex{1}
\latex{1}
\latex{x}
\latex{x}
\latex{y}
\latex{y}
\latex{2a}
\latex{2a}
\latex{3}
\latex{3}
\latex{2a}
\latex{2a}
{{exercise_number}}. Square the following algebraic expressions.
- \latex{ (a-1)^2 }
- \latex{ (x-4)^2 }
- \latex{(x - y)^2}
- \latex{ (1 - z)^2 }
- \latex{ (2x - 1)^2 }
- \latex{ (a-2b)^2}
- \latex{ (4a-3b)^2}
- \latex{ (5x-5y)^2}
{{exercise_number}}. Write algebraic expressions to express the area of the dark green square in different ways.
\latex{x}
\latex{2}
\latex{2}
\latex{x}
\latex{ 3z }
\latex{ y }
\latex{ y }
\latex{ 3z }
\latex{ a }
\latex{ 2b }
\latex{ 2b }
\latex{ a }
{{exercise_number}}. Rewrite the following algebraic expressions using the identities.
- \latex{ (a + 1)^2 }
- \latex{ (y - 2)^2 }
- \latex{(2a - 1)^2 }
- \latex{ (x + 2y)^2 }
- \latex{ (x^2 - 1)^2 }
- \latex{ (a^2 + b^2)^2 }
{{exercise_number}}. Complete the equations so that they are true for all rational numbers.
- \latex{ (x + 2)^2= \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}} + 4x + 4}
- \latex{ (x - 3)^2 = x^2 - \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}} + 9 }
- \latex{(2x - 3y)^2 = 4x^2 - 12xy+ \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}}}
- \latex{ (a^2 + 2b)^2 = a^4 + \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}}+ 4b^2 }
{{exercise_number}}. The sum of the squares of two numbers is \latex{ 125 }. Their difference is \latex{ 5 }. Find their product.
{{exercise_number}}. The sum of two numbers is \latex{ 7 }. Their product is \latex{ 12 }. Find the sum of the squares of the numbers.
{{exercise_number}}. Solve the following equations.
- \latex{ (x + 3)^2 - x^2 = 6 }
- \latex{ (x - 2)^2 = x^2 - 4 }
- \latex{x^2 = (x + 3)^2 - (x + 2) }
- \latex{ (x + 3)^2 - (x - 3)^2 = 18 }
{{exercise_number}}. What remainder does the square of a number leave when divided by \latex{3}, if the number itself leaves a remainder of
- \latex{ 0, }
- \latex{ 1, }
- \latex{ 2 }
when divided by \latex{3?}
{{exercise_number}}. Observe the remainders that square numbers leave when divided by \latex{4.}
{{exercise_number}}. The length of each side of a square is increased by \latex{ 2 } \latex{ cm }. As a result, the area of the square increases by \latex{ 8\ cm^{2} }. Find the length of the sides of the original square.
{{exercise_number}}. The difference of two numbers is \latex{ 3 }; the difference of their squares is \latex{ 15 }. Find the two numbers.
{{exercise_number}}. The flower bed shown in the following image is surrounded by a \latex{ 2 }-\latex{ metre }-wide road. The area of the flower bed is \latex{ 64\ m^{2} } less than the total area, including the road. Find the total area.
\latex{x}
\latex{2}
\latex{2}
\latex{x}
{{exercise_number}}. I thought of a number. The square of the number that is three more than the number I thought of is \latex{ 36 } greater than the square of the number I thought of. Find the number I thought of.
Quiz
The identification number on a credit card consists of \latex{16} digits. The digits from left to right can be expressed as \latex{ a_1; a_2; a_3 ... a_{16}. }
The number
\latex{ 2a_1+a_2+2a_3+a_4+2a_5+a_6+2a_7+a_8+...+2a_{15}+a_{16} } is divisible by \latex{ 10 }.
Two digits are missing from the credit card number:
\latex{ 8543- 09x6-1174-6y38. }
How many different digits can \latex{ x } and \latex{ y } represent?