FORMULA FOR a plus b WHOLE SQUARE
FORMULA FOR a plus b WHOLE SQUARE
In this section, you will learn the formula or expansion for (a + b)2.
That is,
(a + b)2 = (a + b)(a + b)
(a + b)2 = a2 + ab + ab + b2
(a + b)2 = a2 + 2ab + b2
Proving the Expansion of a plus b Whole Square Geometrically
In this section, we are going to see, how to prove the expansion of (a + b)2 geometrically.
We can prove the expansion of (a + b)2 using the area of a square as shown below.
Solved Problems
Problem 1 :
Expand :
(x + y)2
Solution :
(x + y)2 is in the form of (a + b)2
Comparing (a + b)2 and (x + y)2, we get
a = x
b = y
Write the formula / expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute x for a and y for b.
(x + y)2 = x2 + 2(x)(y) + y2
(x + y)2 = x2 + 2xy + y2
So, the expansion of (x + y)2 is
x2 + 2xy + y2
Problem 2 :
Expand :
(x + 2)2
Solution :
(x + 2)2 is in the form of (a + b)2
Comparing (a + b)2 and (x + 2)2, we get
a = x
b = 2
Write the formula / expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute x for a and 2 for b.
(x + 2)2 = x2 + 2(x)(2) + 32
(x + 2)2 = x2 + 4x + 9
So, the expansion of (x + 2)2 is
x2 + 4x + 9
Problem 3 :
Expand :
(5x + 3)2
Solution :
(5x + 3)2 is in the form of (a + b)2
Comparing (a + b)2 and (5x + 3)2, we get
a = 5x
b = 3
Write the expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute 5x for a and 3 for b.
(5x + 3)2 = (5x)2 + 2(5x)(3) + 32
(5x + 3)2 = 25x2 + 30x + 9
So, the expansion of (5x + 3)2 is
25x2 + 30x + 9
Problem 4 :
If a + b = 7 and a2 + b2 = 29, then find the value of ab.
Solution :
To get the value of ab, we can use the formula or expansion of (a + b)2.
Write the formula / expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
or
(a + b)2 = a2 + b2 + 2ab
Substitute 7 for (a + b) and 29 for (a2 + b2).
72 = 29 + 2ab
49 = 29 + 2ab
Subtract 29 from each side.
20 = 2ab
Divide each side by 2.
10 = ab
So, the value of ab is 10.
Problem 5 :
Find the value of :
(√2 + 1/√2)2
Solution :
(√2 + 1/√2)2 is in the form of (a + b)2
Comparing (a + b)2 and (√2 + (1/√2)2, we get
a = √2
b = 1/√2
Write the expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute √2 for a and 1/√2 for b.
(√2 + 1/√2)2 = (√2)2 + 2(√2)(1/√2) + (1/√2)2
(√2 + 1/√2)2 = 2 + 2 + 1/2
(√2 + 1/√2)2 = 9/2
So, the value of (√2 + 1/√2)2 is
9 / 2
Problem 6 :
Find the value of :
(105)2
Solution :
Instead of multiplying 105 by 105 to get the value of (105)2, we can use the algebraic formula for (a + b)2 and find the value of (105)2 easily.
Write (105)2 in the form of (a + b)2.
(105)2 = (100 + 5)2
Write the expansion for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Substitute 100 for a and 5 for b.
(100 + 5)2 = (100)2 + 2(100)(5) + (5)2
(100 + 5)2 = 10000 + 1000 + 25
(105)2 = 11025
So, the value of (105)2 is
11025
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