 Matched Exercise 1:
Find all values of the parameter m in the quadratic equation
x ^{2} + x + m + 1 = 0
such that the equation has
 one solution,
 2 real solutions, and
 2 complex solutions.
Solution to Matched Exercise 1:

Given
x ^{2} + x + m + 1 = 0

Find the discriminant>
D = b^{2}  4ac = 1^{2}  4(1)(m + 1) = 3  4m

For the equation to have one solution, the discriminant has to be equal to zero.
3  4m = 0

Solve the above equation for m.
m = 3/4

For the equation to have 2 real solution, the discriminant has to be greater than zero.
3  4m > 0

The inequality 3  4m > 0 has the following solution set.
(∞ ,  3/4)

For the equation to have 2 complex solution, the discriminant has to be less than zero.
3  4m < 0

The inequality 3  4m > 0 has the following solution set.
( 3/4 , + ∞)
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