math problem polynomials?

Question

what number must be added to 9x^2 + 9x to form a perfect square? please show work/ explain it.

Answer 1

9x^2 + 9x
= 9(x^2 +x)
= 9(x^2 +x +1/4) -9/4
So you must add 9/4

Answer 2

When I was taking these classes, we were taught a simple way to expand squared binomials into a polynomial.

For instance if it was (a+b)^2, it would be a^2 + 2ab + b^2

With this in mind, you need to work backwards. You know that a=3 in the squared binomial. So you know that 2*3*x=9(the b value). So what you want to do is divide: 9/6 = 1.5

so the answer is (3x + 1.5)^2 Which turns the number that must be added to 1.5^2, or 2.25.

Hope it wasn't too confusing, and I hope it helped.

Answer 3

havent done these in awhile so please verify before taking my word...

a) 9x^2 + 9x + c
b) (3x + t)(3x+t) = 9x^2+9x+c ---> use FOIL --->form of perfect square
c) 9x^2 + 6xt + t^2 = 9x^2 + 9x + c ----> just expanded
d) t^2 + 6xt = 9x +c ------> combine like terms
e) 6xt + t^2 = 9x ------> solve for t
f) 6t = 9
g) t = 9/6 = 3/2

so t = 3/2

Verification:
* (3x+t) (3x+t) and t = 3/2
a) (3x + 3/2) (3x + 3/2)
= 9x^2 +((9/2)x + (9/2)x) + 9/4
= 9x^2 + (18/2)x + 9/4
= 9x^2 + 9x + 9/4

so c = 9/4