should be added to ${4 x}^{2} - 4 x$ to make it a perfect square.

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Solution

The correct option is A 1
Let us compare ${4 x}^{2} - 4 x$ with expansion of ${(a + b)}^{2}$ .

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

On comparing terms of RHS of above equation and $4 x^{2} - 4 x$ , it is clear that-

$a^{2} = 4 x^{2}$ .... (1)

$- 4 x = 2 a b$ .... (2)

From (1) $a = 2 x$

Substituting value of a in (2),

$- 4 x = 2 \times (2 x) \times b$

$\Rightarrow b = - 1$

Therefore, $b^{2}$ is the term that needs to be added to it to make it perfect square.

Thus, we need to add 1.

Therefore, if we add 1 to ${4 x}^{2} - 4 x$ , it becomes

${4 x}^{2} - 4 x + 1 = (2 x - 1)^{2}$

which is a perfect square.

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