Question

Use identities to solve $ {(97)^2} $
A.9,695
B.9,409
C.9,009
D.9,209

Answer 1

Hint: We need to convert the above equation in the form of $ {(a + b)^2} $ or $ {(a - b)^2} $ based on ease of solving and then finding the value accordingly using the formulas.
Formula:
 $ {(a - b)^2} = {a^2} + {b^2} - 2ab $
 $ {(a + b)^2} = {a^2} + {b^2} + 2ab $

Complete step-by-step answer:
First, we have to understand the meaning of identity equation which we will use to solve the equation
An identity equation is an equation that is always true for any value substituted into the variable i.e. no matter whatsoever value we put at the place of the variables of the equation the algebraic equation remains valid. They are also used for the factorization of polynomials including computation of algebraic expressions and solving different polynomials.
Now, to solve the above equation we need to express the equation $ {(97)^2} $ in the form of $ \Rightarrow {(a - b)^2} $ or $ {(a + b)^2} $
Hence, $ {(97)^2} $ can be written as $ {(100 - 3)^2} $ or $ {(90 + 7)^2} $
From the above two forms we need to choose any one based on our convenience. According to me, $ {(100 - 3)^2} $ will be comparatively easier so let go with it.
So, $ {(97)^2} = {(100 - 3)^2} $
 $ = {(100)^2} + {(3)^2} - (2 \times 100 \times 3) $ [using the formula $ {(a - b)^2} = {a^2} + {b^2} - 2ab $ ]
 $ = 10000 + 9 - 600 $
 $ = 9409 $
Hence, the value of $ {(97)^2} $ is $9409$.
So, the correct answer is “Option B”.

Additional Information:
The above two formulas are the most frequently used equations. Similarly, we have few more basic identity equations which we need to memorize for lifelong because we may come across them frequently. Kindly go through the below equation and try to memorize them.
Identity I: $ {(a + b)^2} = {a^2} + {b^2} + 2ab $
Identity II: $ {(a - b)^2} = {a^2} + {b^2} - 2ab $
Identity III: $ {a^2} - {b^2} = (a + b)(a - b) $
Identity IV: $ (x + a)(x + b) = {x^2} + (a + b)x + ab $
Identity V: $ {(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca $
Identity VI: $ {(a + b)^3} = {a^3} + {b^3} + 3ab(a + b) $
Identity VII: $ {(a - b)^3} = {a^3} - {b^3} - 3ab(a - b) $
Identity VIII: $ {a^3} + {b^3} + {c^3} - 3abc = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ca) $

Note: We need to choose in between $ {(a - b)^2} $ and $ {(a + b)^2} $ based on our time availability and comfort level. Both the equations will eventually land you to the right answer. Additionally, we should try to remember the above formulas for solving the equations conveniently.