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16 Mar 2021, 01:30
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C
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Difficulty:
65%
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Question Stats:
52% (01:46) correct
48%
(01:56)
wrong
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If \(x^2 + 12x − k = 0\), is \(x = 4\)?
(1) \((x + 16)\) is a factor of \(x^2 + 12x − k\), where k is a constant, and x is a variable.
(2) \(x ≠ −16\)
(1) \((x + 16)\) is a factor of \(x^2 + 12x − k\), where k is a constant, and x is a variable.
(2) \(x ≠ −16\)
17 Mar 2021, 01:15
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Bunuel
If \(x^2 + 12x − k = 0\), is \(x = 4\)?
(1) \((x + 16)\) is a factor of \(x^2 + 12x − k = 0\), where k is a constant, and x is a variable.
(2) \(x ≠ −16\)
(1) \((x + 16)\) is a factor of \(x^2 + 12x − k = 0\), where k is a constant, and x is a variable.
(2) \(x ≠ −16\)
If x^2+12x−k=0, is x=4?
(1) (x+16) is a factor of x^2+12x−k=0, where k is a constant, and x is a variable. This implies that we can factor out \(x+16\) from \(x^2+12x-k=0\), so we would have \((x+16)*(something)=0\). Thus x=-16 is one of the roots of the given quadratic equation.
Viete's theorem states that for the roots \(x_1\) and \(x_2\) of a quadratic equation \(ax^2+bx+c=0\):
\(x_1+x_2=\frac{-b}{a}\) AND \(x_1*x_2=\frac{c}{a}\).
Thus according to the above \(x_1+x_2=-16+x_2=\frac{-12}{1}\) --> \(x_2=4\).
So, we have that x is either -16 or 4. Not sufficient.
(2) x≠−16. Clearly insufficient.
(1)+(2) Since (2) says that x is NOT -16, then x=4. Sufficient.
Answer: C.
Hope it's clear.
General Discussion
16 Mar 2021, 02:50
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Bunuel
If \(x^2 + 12x − k = 0\), is \(x = 4\)?
(1) \((x + 16)\) is a factor of \(x^2 + 12x − k\), where k is a constant, and x is a variable.
(2) \(x ≠ −16\)
(1) \((x + 16)\) is a factor of \(x^2 + 12x − k\), where k is a constant, and x is a variable.
(2) \(x ≠ −16\)
x^2 + 12x − k = 0
Sum of roots = -12
Product of roots = -k
From statement 1:
(x + 16) is a factor => x = -16 is a root
Since sum of roots is -12, the other root must be 4
Thus:
x^2 + 12x − k = (x - 4)(x + 16) = 0
=> x = 4 or 16
Thus, x may be 4 or 16 - insufficient
Alternative approach:
x = -16 is a root
Substitute x = -16 in given equation:
(-16)² + 12(-16) - k = 0
k = 256 - 192 = 64
x^2 + 12x − k
x^2 + 12x − 64 = (x - 4)(x + 16) = 0
=> x = 4 or 16
Thus, x may be 4 or 16 - insufficient
From statement 2:
Clearly this is insufficient since we don't know the value of k
Combining:
x = 4 (since it is not -16)
Sufficient
Answer C
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