Sure, let's solve the equation [tex]\((2x + 3y)^2 = (a^2 x + b y^2)^2\)[/tex] step-by-step.
Step 1: Expand the left-hand side
We start with the expression [tex]\((2x + 3y)^2\)[/tex]. Using the binomial theorem or distributive property [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], we get:
[tex]\[
(2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2
\][/tex]
Calculating each term individually:
[tex]\[
(2x)^2 = 4x^2
\][/tex]
[tex]\[
2(2x)(3y) = 12xy
\][/tex]
[tex]\[
(3y)^2 = 9y^2
\][/tex]
Adding these results together gives the expanded left-hand side:
[tex]\[
4x^2 + 12xy + 9y^2
\][/tex]
Step 2: Expand the right-hand side
Next, we expand the expression [tex]\((a^2 x + b y^2)^2\)[/tex]. Similarly, using the binomial theorem or distributive property, we get:
[tex]\[
(a^2 x + b y^2)^2 = (a^2 x)^2 + 2(a^2 x)(b y^2) + (b y^2)^2
\][/tex]
Calculating each term individually:
[tex]\[
(a^2 x)^2 = a^4 x^2
\][/tex]
[tex]\[
2(a^2 x)(b y^2) = 2a^2 b x y^2
\][/tex]
[tex]\[
(b y^2)^2 = b^2 y^4
\][/tex]
Adding these results together gives the expanded right-hand side:
[tex]\[
a^4 x^2 + 2a^2 b x y^2 + b^2 y^4
\][/tex]
Step 3: Write the expanded equations
The expanded form of the original equation [tex]\((2x + 3y)^2 = (a^2 x + b y^2)^2\)[/tex] is:
[tex]\[
4x^2 + 12xy + 9y^2 = a^4 x^2 + 2a^2 b x y^2 + b^2 y^4
\][/tex]
So, the final result of expanding both sides of the equation [tex]\((2x + 3y)^2 = (a^2 x + b y^2)^2\)[/tex] is:
[tex]\[
4x^2 + 12xy + 9y^2 = a^4 x^2 + 2a^2 b x y^2 + b^2 y^4
\][/tex]
