Answer:
k = 6
Step-by-step explanation:
To make the expression x² + kxy + 9y² a perfect square, we want it to be in the form (a + b)² for some values of a and b.
Expanding (a + b)², we get:
[tex](a+b)^2=a^2+2ab+b^2[/tex]
Now, we can compare each term of the given expression with the expanded form of a perfect square:
[tex]x^2+kxy+9y^2=a^2+2ab+b^2[/tex]
Compare the first terms:
[tex]x^2=a^2 \implies x=a[/tex]
Compare the last terms:
[tex]9y^2=b^2\implies (3y)^2=b^2 \implies b=3y[/tex]
Compare the middle terms:
[tex]kxy=2ab[/tex]
Substitute in a = x and b = 3y:
[tex]kxy=2(x)(3y)[/tex]
[tex]kxy=6xy[/tex]
[tex]k=6[/tex]
Therefore, the expression x² + kxy + 9y² becomes a perfect square when:
[tex]\Large\boxed{\boxed{k=6}}[/tex]
