The solutions of the quadratic equation y^2 + 6y + 9 = 0 can easily be found without finding the discriminant.
y^2 + 6y + 9 = 0
=> y^2 + 3y + 3y + 9 = 0
=> y( y + 3) + 3(y + 3) = 0
=> (y + 3)^2 = 0
=> y = -3
Or you can use the formula (a + b)^2 = a^2 + b^2 + 2ab.
The solution of the quadratic equation obtained without finding the discriminant is -3
Well, it is simple, since we recognize in the given form of the quadratic a perfect square.
We've had been driven by the formula:
(a+b)^2 = a^2 + 2ab + b^2
If we'll put a^2 = y^2 and b^2 = 9 => b=3 and a=y
2ab = 2*3*y = 6y
y^2+6y+9 = (y+3)^2
We'll solve the quadratic:
(y+3)^2 = 0
(y+3)(y+3) = 0
We'll put y+3=0
The solutions of the quadratic are real numbers and they are equal: y1=y2 = -3.