Question

Simplify using appropriate identities.

(3.5p-2.5q)2

- (2.5p-3.5q)2​

Answer 1

On simplification, $(3.5p-2.5q)^{2} -(2.5p-3.5q)^{2}=6p^{2} -6q^{2}$

To simplify $(3.5p-2.5q)^{2} -(2.5p-3.5q)^{2}$,

By using the identity $(a-b)^{2}=a^{2}-2ab+b^{2}$,

We get

$(3.5p-2.5q)^{2}=(3.5p)^{2} -2*3.5p*2.5q+(2.5q)^{2}\\ =12.25p^{2}-17.5pq+6.25\\\\(2.5p-3.5q)^{2}=(2.5p)^{2} -2*2.5p*3.5q+(3.5q)^{2} \\=6.25p^{2} -17.5pq+12.25q^{2}$

Then,

$(3.5p-2.5q)^{2}-(2.5p-3.5q)^{2}=12.25p^{2}-17.5pq+6.25q^{2} -(6.25p^{2}17.5pq+12.25q^{2})\\\\=12.25p^{2}-17.5pq+6.25q^{2}-6.25p^{2} +17.5pq-12.25q^{2}\\\\=12.25p^{2}-6.25p^{2}-17.5pq +17.5pq+6.25q^{2}-12.25q^{2}\\\\=6p^{2}-6q^{2}$

Some of similar useful identities are,

$(a+b)^{2} =a^{2}+2ab+b^{2} \\a^{2} -b^{2} =(a+b)(a-b)\\(x+a)(x+b)=x^{2} +(a+b)x+ab\\(a+b+c)^{2}=a^{2} +b^{2} +c^{2}+2ab+2bc+ac$

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