The part of the quadratic formula that determines the the number of solutions to an equation is the part under the square root sign: b2 - 4ac (the discriminant).
- When b2 - 4ac > 0, there are two real & distinct solutions (crosses x-axis at two points).
- When b2 - 4ac = 0, there is one, repeated solution (just touches x-axis at one point).
- When b2 - 4ac < 0, there are no real solutions (doesn't cross x-axis).
Examples: 1. Find the discriminant & solve the equation 4x2 + 12x + 9 = 0
discriminant = b2 - 4ac = 122 - 4 x 4 x 9 = 0 b2 - 4ac = 0, hence one solution x = -3/2
2. Find the value of the constant p such that he equation x2 + x + p has equal roots.
a = 1, b = 1, c = p b2 - 4ac = 0 12 - 4 x 1 x p = 0 1 - 4p = 0 1 = 4p p = 1/4
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