(ax+by)² + (bx+ay)²
Answers
Step-by-step explanation:
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
)
Hence the given sum is factorized