Mo2men wrote:If k is an integer and x(x - k) = k + 1, what is the value of x?
(1) x < k
(2) x = 3 - k
Statement 1: x<k
Case 1: k=0
Substituting k=0 into x(x - k) = k + 1, we get:
x(x - 0) = 0 + 1
x² = 1
x = ±1.
Since statement 1 requires that x<k, the only viable option is x=-1.
Case 2: k=10
Substituting k=10 into x(x - k) = k + 1, we get:
x(x - 10) = 10 + 1
x² - 10x = 11
x² - 10x - 11 = 0
(x-11)(x+1) = 0
x = 11 or x=-1.
Since statement 1 requires that x<k, the only viable option is x=-1.
Case 3: k=-1
Substituting k=-1 into x(x - k) = k + 1, we get:
x(x + 1) = -1 + 1
x² + x = 0
x = 0.
Since statement 1 requires that x<k, this case is not viable.
Case 4: k=-10
Substituting k=-10 into x(x - k) = k + 1, we get:
x(x + 10) = -10 + 1
x² + 10x = -9
x² + 10x + 9 = 0
(x+9)(x+1) = 0
x = -9 or x=-1.
Since neither option for x is such that x<k, this case is not viable.
In every viable case, x=-1.
SUFFICIENT.
Statement 2: x=3-k
Substituting k = 3-x into x(x - k) = k + 1, we get:
x(x - (3-x)) = 3-x + 1
x(2x - 3) = 4 - x
2x² - 3x = 4 - x
2x² - 2x - 4 = 0
x² - x - 2 = 0
(x-2)(x+1) = 0
x=2 or x=-1.
Since x can be different values, INSUFFICIENT.
The correct answer is
A.
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