Question

For what value of p and q is y^4 + py^3 + qy^2 + 12y + 9 a perfect square ? ​

Answer 1

Given :- For what value of p and q is y^4 + py^3 + qy^2 + 12y + 9 a perfect square ?

Answer :-

→ y^4 + py^3 + qy^2 + 12y + 9

we know that,

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

→ (y²)² + p * y² * y + q * (y)² + 2 * 3 * (2y) + (3)²

→ (y²)² + (3)² + q *(y)² + p* y * y² + 2 * 3 * (2y)

comparing we get,

• a = y²
• b = 3
• c = 2y { 2 * 3 * 2y }

then,

→ (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

→ (y²)² + (3)² + (2y)² + 2*y²*3 + 2*3*2y + 2 * 2y * y²

→ (y²)² + (3)² + (2y)² + 6y² + 12y + 4y³

→ (y²)² + (3)² + 4y² + 6y² + 12y + 4y³

→ y⁴ + 9 + 10y² + 12y + 4y³

→ y⁴ + 4y³ + 10y² + 12y + 9 .

therefore,

• p = 4
• q = 10

Learn more :-

JEE mains Question :-

https://brainly.in/question/22246812

Answer 2

$\textbf{Given:}$

$\mathsf{y^4+p\,y^3+q\,y^2+12y+9\;is\;a\;perfect\;square}$

$\textbf{To find:}$

$\textsf{The values of p and q}$

$\textbf{Solution:}$

$\textsf{Consider,}$

$\mathsf{y^4+p\,y^3+q\,y^2+12y+9}$

$\textsf{This can be written as}$

$\mathsf{y^4+p\,y^3+q\,y^2+12y+9=(y^2+m\,y+3)^2}$

$\mathsf{y^4+p\,y^3+q\,y^2+12y+9=(y^2+m\,y+3)\,(y^2+m\,y+3)}$

$\underline{\textsf{Equating coefficient of y:}}$

$\mathsf{12=3m+3m}$

$\mathsf{12=6\,m}$

$\implies\boxed{\mathsf{m=2}}$

$\underline{\mathsf{Equating\;coefficient\;of \;y^2:}}$

$\mathsf{q=6+m^2}$

$\mathsf{q=6+2^2}$

$\mathsf{q=6+4}$

$\implies\boxed{\mathsf{q=10}}$

$\underline{\mathsf{Equating\;coefficient\;of \;y^3:}}$

$\mathsf{p=m+m}$

$\mathsf{p=2+2}$

$\mathsf{p=4}$

$\implies\boxed{\mathsf{p=4}}$

$\textbf{Find more:}$