To determine which equation has exactly one solution, we need to analyze each option. We will set them equal to zero and check their forms.
A
Given: \[ 10x^2 - 4x - 23 = 0 \]
The discriminant (D) of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[ D = b^2 - 4ac \]
For \(10x^2 - 4x - 23\):
- \(a = 10\)
- \(b = -4\)
- \(c = -23\)
Calculating the discriminant: \[ D = (-4)^2 - 4(10)(-23) = 16 + 920 = 936 \] A positive discriminant indicates two real solutions.
B
Given: \[ 5x + 4 - 415x + 5 - 3 = 0 \] This simplifies to: \[ -410x + 6 = 0 \] This is a linear equation which has exactly one solution.
C
Given: \[ 6x^2 - 3x - 12 = 0 \]
Calculating the discriminant:
- \(a = 6\)
- \(b = -3\)
- \(c = -12\)
\[ D = (-3)^2 - 4(6)(-12) = 9 + 288 = 297 \] This also has a positive discriminant, indicating two real solutions.
D
Given: \[ 2x - 5 = 0 \] This is a linear equation which has exactly one solution.
Conclusion
Both options B and D are linear equations that have exactly one solution.
If asked for just one option with exactly one solution, you could say either option depending on the context. Generally, if only the distinct form of the equation is needed, then:
- B could be the best choice based on the context of the equation being a well-defined linear one.
A, C have multiple solutions, while B and D have exactly one:
B or D has exactly one solution.