 UPSKILL MATH PLUS

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Identity –$$3$$: ${\left(a-b\right)}^{2}={a}^{2}-\mathit{2ab}+{b}^{2}$

If we replace the variable $$(-b)$$ in the identity -$$(2)$$ instead of $$(b)$$, we get a new identity, that is${\left(a-b\right)}^{2}={a}^{2}-\mathit{2ab}+{b}^{2}$.

Let us see how we got it.

We know the identity –$$(2)$$, that is ${\left(a+b\right)}^{2}={a}^{2}+\mathit{2ab}+{b}^{2}$. Substitute the variable $$(-b)$$ in the identity $$(2)$$ instead of $$(b)$$.

Then, we get:

$\begin{array}{l}{\left(a+\left(-b\right)}^{2}=\left(a+\left(-b\right)\right)\left(a+\left(-b\right)\right)\\ \\ ={a}^{2}+a\left(-b\right)+\left(-b\right)a+{\left(-b\right)}^{2}\\ \\ {\left(a-b\right)}^{2}={a}^{2}+2a\left(-b\right)+\left(-b\right)×\left(-b\right)\\ \\ {\left(a-b\right)}^{2}={a}^{2}-2\mathit{ab}+{\left(-b\right)}^{2}\end{array}$

From the above equation, we can observe that when we change the sign of $$b$$, the second term $$2ab$$ alone is changed, the other term remains unchanged.

Now we see an example to understand this concept better.
Example:
Simplify the expression ${\left(4x-11y\right)}^{2}$ using the appropriate identity.

We can simplify the given expression ${\left(4x-11y\right)}^{2}$, using the identity ${\left(a-b\right)}^{2}={a}^{2}-\mathit{2ab}+{b}^{2}$.

Here, $$a =4x$$; $$b = 11y$$

Now, substitute $$a$$, and $$b$$ values in the identity.

$\begin{array}{l}={\left(4x\right)}^{2}-2\left(4x×11y\right)+{\left(-11y\right)}^{2}\\ \\ =16{x}^{2}-2\left(44xy\right)+121{y}^{2}\\ \\ =16{x}^{2}-88xy+121{y}^{2}\end{array}$

Therefore, ${\left(4x-11y\right)}^{2}$ $$=$$ $16{x}^{2}-88xy+121{y}^{2}$.