Identifying the Equation with a Single Solution
To determine which of the given equations has only one solution, we need to analyze each equation individually. The main focus will be on the nature of their roots, which can be discerned by examining their discriminants.
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Equation: x² – 4x + 4 = 0
The discriminant (D) of a quadratic equation ax² + bx + c is given by the formula:
D = b² – 4ac
In this case, a = 1, b = -4, and c = 4.
Calculating the discriminant:
D = (-4)² – 4(1)(4) = 16 – 16 = 0
Since the discriminant is 0, this equation has exactly one solution (also known as a double root).
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Equation: x² + x = 0
Rearranging gives us x(x + 1) = 0. The solutions are found by setting each factor to zero:
x = 0 or x + 1 = 0 => x = -1
This equation has two distinct solutions: x = 0 and x = -1.
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Equation: x² – 1 = 0
This can be factored as (x – 1)(x + 1) = 0. Solving this gives:
x – 1 = 0 => x = 1 or x + 1 = 0 => x = -1
This equation also has two distinct solutions: x = 1 and x = -1.
Conclusion
Out of the three equations presented, the equation that has only one solution is:
x² – 4x + 4 = 0
This conclusion is based on its discriminant being zero, which indicates that it has a double root at x = 2.