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Express the algebraic expression in a simplified manner: $ 4{a^2}{b^2} - 12abc + 9{c^2} $ .

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: The given algebraic expression is a complicated expression generated by a simple expression that is binomial expansion of $ {(x - y)^2} $ . We will use the formula of this expression and try to simplify the given expression in the shortest form possible.
Formula used:
The binomial expansion for the expression $ {(x - y)^2} $ is given by
 $ {(x - y)^2} = {x^2} - 2xy + {y^2} $

Complete step-by-step answer:
The given algebraic expression is:
 $ 4{a^2}{b^2} - 12abc + 9{c^2} $
The given equation looks similar to the expression \[{x^2} - 2xy + {y^2}\]
Comparing the given equation to this expression we get:
 $\Rightarrow {x^2} = 4{a^2}{b^2}, - 2xy = - 12abc,{y^2} = 9{c^2} $
On solving each thing individually we have:
 $ {x^2} = 4{a^2}{b^2} \Rightarrow x = \sqrt {4{a^2}{b^2}} = \sqrt {{{(2ab)}^2}} = 2ab $ --(1)
 $ - 2xy = - 12abc = - 2 \times (2ab) \times (3c) $ --(2)
 $ {y^2} = 9{c^2} \Rightarrow y = \sqrt {9{c^2}} = \sqrt {{{(3c)}^2}} = 3c $ --(3)
Since, from the binomial expansion of $ {(x - y)^2} $ we have:
\[{x^2} - 2xy + {y^2} = {(x - y)^2}\]
Putting the values obtained in (1),(2) and (3) we get
 $ 4{a^2}{b^2} - 12ab + 9{c^2} = {(2ab)^2} - 2 \times (2ab) \times (3c) + {(3c)^2} $
 $ \Rightarrow 4{a^2}{b^2} - 12ab + 9{c^2} = {(2ab - 3c)^2} $
Therefore, the simplification of the algebraic expression $ 4{a^2}{b^2} - 12abc + 9{c^2} $ gives us $ {(2ab - 3c)^2} $ .
So, the correct answer is “ $ {(2ab - 3c)^2} $ ”.

Additional information:
There are various formulas one should remember to make simplification easier such as:
1) $ {(x - y)^2} = {x^2} - 2xy + {y^2} $
2) $ {(x + y)^2} = {x^2} + 2xy + {y^2} $
3) $ (x - y)(x + y) = {x^2} - {y^2} $
With these formulas in mind, all algebraic expressions can be simplified and expanded as well. These formulas can also be verified. These are called the special binomial formulas. This formula was given by Greek mathematician Euclid, later it was generalized to $ {(x - y)^n} $ . By simple multiplication we can prove that $ {(x + y)^2} = {x^2} + 2xy + {y^2} $ and the other formulas too.

Note: Looking at the question first see all the possibilities to find the simplified answer. Think what formulas you are going to use. If you are not clear about the expression first try to split it into various terms, it will always give you an idea of factoring the expression.
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