Math, asked by farhin54, 2 months ago

(ax + by)2 + (bx - ay)2​

Answers

Answered by goswamirudransh72
0

Answer:

Factorization is the decomposition of mathematical objects into the product of smaller or simpler objects

Factorizing helps in finding the roots of the factors

Consider the given equation

=(a x+b y)^{2}+(b x-a y)^{2}=(ax+by)

2

+(bx−ay)

2

By applying (a+b)^{2}(a+b)

2

and (a-b)^{2}(a−b)

2

formula for the above equation

We get

=\left[a^{2} x^{2}+b^{2} y^{2}+2 a b x y\right]+\left[b^{2} x^{2}+a^{2} y^{2}-2 a b x y\right]=[a

2

x

2

+b

2

y

2

+2abxy]+[b

2

x

2

+a

2

y

2

−2abxy]

Simplify the above equation

\begin{gathered}\begin{array}{l}{=a^{2} x^{2}+b^{2} y^{2}+2 a b x y+b^{2} x^{2}+a^{2} y^{2}-2 a b x y} \\ {=a^{2} x^{2}+b^{2} y^{2}+b^{2} x^{2}+a^{2} y^{2}}\end{array}\end{gathered}

=a

2

x

2

+b

2

y

2

+2abxy+b

2

x

2

+a

2

y

2

−2abxy

=a

2

x

2

+b

2

y

2

+b

2

x

2

+a

2

y

2

By taking the common terms

We get the above equation as

=a^{2}\left(x^{2}+y^{2}\right)+b^{2}\left(x^{2}+y^{2}\right)=a

2

(x

2

+y

2

)+b

2

(x

2

+y

2

)

Here \left(x^{2}+y^{2}\right)(x

2

+y

2

) is written only once and \left(a^{2}+b^{2}\right)(a

2

+b

2

) is combined

=\left(x^{2}+y^{2}\right)\left(a^{2}+b^{2}\right)=(x

2

+y

2

)(a

2

+b

2

)

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