In this section, we are going to see the formula or factored form of (a2 - b2)
That is,
a2 - b2 = (a + b)(a - b)
Problem 1 :
Factor :
x2 - y2
Solution :
x2 - y2 is in the form of a2 - b2
Comparing a2 - b2 and x2 - y2, we get
a = x
b = y
Write the formula for a2 - b2.
a2 - b2 = (a + b)(a - b)
Substitute x for a and y for b.
x2 - y2 = (x + y)(x - y)
So, the factors of x2 - y2 are
(x + y) and (x - y)
Problem 2 :
Factor :
x2 - 4
Solution :
(x2 - 4) can be written as
x2 - 22
x2 - 22 is in the form of a2 - b2
Comparing a2 - b2 and x2 - 22, we get
a = x
b = 2
Write the formula for a2 - b2.
a2 - b2 = (a + b)(a - b)
Substitute x for a and 2 for b.
x2 - 22 = (x + 2)(x - 2)
So, the factors of x2 - 4 are
(x + 2) and (x - 2)
Problem 3 :
Factor :
25x2 - 9
Solution :
(25x2 - 9) can be written as
(5x)2 - 32
(5x)2 - 32 is in the form of a2 - b2
Comparing a2 - b2 and (5x)2 - 32, we get
a = 5x
b = 3
Write the formula for a2 - b2.
a2 - b2 = (a + b)(a - b)
Substitute 5x for a and 3 for b.
(5x)2 - 32 = (5x + 3)(5x - 3)
So, the factors of 25x2 - 9 are
(5x + 3) and (5x - 3)
Problem 4 :
If x2 - y2 = 16 and x + y = 8, then find the value of
(x - y)
Solution :
We can factor (x2 - y2) using the formula
(a2 - b2) = (a + b)(a - b)
That is
x2 - y2 = (x + y)(x - y)
Substitute 16 for (x2 - y2) and 8 for (x + y).
16 = 8(x - y)
Divide each side by 8.
2 = x - y
So, the value of (x - y) is 2.
Problem 5 :
If 36x2 - 9y2 = 52 and 6x - 3y = 4, then find the value of
(6x + 3y)
Solution :
We can factor (36x2 - 9y2) using the formula
(a2 - b2) = (a + b)(a - b)
That is
36x2 - 9y2 = (6x)2 - (3y)2
36x2 - 9y2 = (6x + 3y)(6x - 3y)
Substitute 52 for (36x2 - 9y2) and 4 for (6x - 3y).
52 = (6x + 3y) ⋅ 4
Divide each side by 4.
13 = 6x + 3y
So, the value of (6x + 3y) is 13.
Problem 6 :
Find the value of the numerical expression given below using algebraic identity.
(12)2 / 96
Solution :
(12)2 / 96 = (12)2 / (100 - 4)
(12)2 / 96 = (12)2 / (102 - 22)
Factor (102 - 22) using the formula for (a2 - b2).
(12)2 / 96 = (12)2 / [(10 + 2)(10 - 2)]
(12)2 / 96 = (12)2 / [(12)(8)]
(12)2 / 96 = 12 / 8
(12)2 / 96 = 3 / 2
So, the value of (12)2 / 96 is
3/2
Algebraic identities are equalities which remain true regardless of the values of any variables which appear within it.
To know more identities in Algebra,
In our website, we have provided two calculators for algebra identities.
One is to find the expansion for (a + b)n and other one is to find the expansion for (a - b)n.
Please click the below links to get expansion calculator that you need.
Expansion Calculator for (a + b)n
Expansion Calculator for (a - b)n
If you would like to have problems on algebraic identities, please click the link given below.
Worksheet on Algebraic Identities
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