Determine an ordered pair $$ (x ; y) $$ which satisfies the equation. $$ (2 q+x)^{2}=4 q^{2}+12 q+y $$

Question

UpStudy AI · Accepted Answer

To solve the equation

$$
(2q + x)^2 = 4q^2 + 12q + y,
$$

we start by expanding the left-hand side:

$$
(2q + x)^2 = (2q)^2 + 2 \cdot (2q) \cdot x + x^2 = 4q^2 + 4xq + x^2.
$$

Now, equate the expanded form to the right-hand side:

$$
4q^2 + 4xq + x^2 = 4q^2 + 12q + y.
$$

Since this equation must hold for all values of $$ q $$, we compare the coefficients of like terms.

1. Coefficient of $$ q^2 $$:
   $$
   4 = 4.
   $$
   This holds true.

2. Coefficient of $$ q $$:
   $$
   4x = 12 \quad \Rightarrow \quad x = 3.
   $$

3. Constant term:
   $$
   x^2 = y \quad \Rightarrow \quad y = 3^2 = 9.
   $$

Thus, the ordered pair that satisfies the equation is

$$
(x; y) = (3; 9).
$$
The ordered pair that satisfies the equation is $$ (3, 9) $$.

Determine an ordered pair \( (x ; y) \) which satisfies the equation. \( (2 q+x)^{2}=4 q^{2}+12 q+y \)