Step-by-step explanation:
For a trinomial ax² + bx + c = 0, the discriminant is b² − 4ac.
If the discriminant is positive, there are 2 real solutions.
If the discriminant is 0, there is 1 real solution.
If the discriminant is negative, there are no real solutions.
a) x² − 4x − 7 = 0
Here, a = 1, b = -4, and c = -7.
b² − 4ac = (-4)² − 4(1)(-7) = 44
The discriminant is positive, so there are 2 real solutions.
b) 4r² + 11r − 3 = 0
Here, a = 4, b = 11, and c = -3.
b² − 4ac = (4)² − 4(11)(-3) = 148
The discriminant is positive, so there are 2 real solutions.
c) 3m² + 7 = 0
Here, a = 3, b = 0, and c = 7.
b² − 4ac = (0)² − 4(3)(7) = -84
The discriminant is negative, so there are no real solutions.
d) t² + 2t + 1 = 0
Here, a = 1, b = 2, and c = 1.
b² − 4ac = (2)² − 4(1)(1) = 0
The discriminant is zero, so there is 1 real solution.
