To determine which equation has exactly one real solution, we can use the concept of the discriminant from the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / 2a
The discriminant, which is the part b² – 4ac, plays a crucial role in identifying the nature of the roots:
- If the discriminant is greater than zero, the equation has two distinct real solutions.
- If the discriminant is equal to zero, the equation has exactly one real solution (a repeated root).
- If the discriminant is less than zero, the equation has no real solutions (the roots are complex).
Let’s analyze each of the given equations:
- a) 4x² – 12x + 9 = 0
Here, a = 4, b = -12, and c = 9.
Discriminant = (-12)² – 4(4)(9) = 144 – 144 = 0.
This equation has exactly one real solution. - b) 4x² + 12x + 9 = 0
Here, a = 4, b = 12, and c = 9.
Discriminant = (12)² – 4(4)(9) = 144 – 144 = 0.
This equation also has exactly one real solution. - c) 4x² – 6x + 9 = 0
Here, a = 4, b = -6, and c = 9.
Discriminant = (-6)² – 4(4)(9) = 36 – 144 = -108.
This equation has no real solutions. - d) 4x² + 6x + 9 = 0
Here, a = 4, b = 6, and c = 9.
Discriminant = (6)² – 4(4)(9) = 36 – 144 = -108.
This equation also has no real solutions.
From the analysis, both equations a) and b) have exactly one real solution. Therefore, the answer to the question is:
Answer: a) 4x² – 12x + 9 = 0 and b) 4x² + 12x + 9 = 0 both have exactly one real solution.