Rule:

Ax² + Bx + C = 0

1. Substitute A, B, and C in the discriminant formula

       Discriminant = B² - 4AC

2. If the discriminant is a positive number then the equation
   has two real solutions.

3. If the discriminant is a negative number then the equation
   has two conjugate imaginary solutions.
  
4. If the discriminant is zero then the equation
   has one real solution.

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x² - 4x - 5 = 0

A = 1, B = -4, C = -5

Substitute A=1, B=-4, and C=-5 in the discriminant formula

       Discriminant = B² - 4AC
       Discriminant = (-4)² - 4(1)(-5)
       Discriminant = 16 + 20
       Discriminant = 36

36 is a negative number, so the equation has two real 
solutions.

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x² - 12x + 36 = 0

A = 1, B = -12, C = 36

Substitute A=1, B=-12, and C=36 in the discriminant formula

       Discriminant = B² - 4AC
       Discriminant = (-12)² - 4(1)(36)
       Discriminant = 144 - 144
       Discriminant = 0

That came out zero, so the equation has one 
real solutions.

Edwin