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What is the number of roots of the equation (x^2-5x+5)^{(x^2

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by Max@Math Revolution » Mon Apr 01, 2019 1:50 am

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[GMAT math practice question]

What is the number of roots of the equation (x^2-5x+5)^{(x^2-5x-6)}=1?

A. 2
B. 3
C. 4
D. 5
E. 6
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by Max@Math Revolution » Wed Apr 03, 2019 2:54 am
=>

When we have an equation of the form a^b = 1, we need to consider the following three cases:
Case 1: a = 1
Case 2: a = -1 and b is an even number
Case 3: b = 0.

Case 1: x^2-5x+5 = 1
=> x^2-5x+4 = 0
=> (x-1)(x-4) = 0
=> x = 1 or x = 4
Thus, 1 and 4 are roots of the equation.

Case 2: x^2-5x+5 = -1
=> x^2-5x+6 = 0
=> (x-2)(x-3) = 0
=> x = 2 or x = 3
If x = 2, then the exponent b = x^2-5x-6 = 4 - 10 - 6 = -12 is an even number.
If x = 3, then the exponent b = x^2-5x-6 = 9 - 15 - 6 = -12 is an even number.
Thus, 2 and 3 are roots of the equation.

Case 3: x^2-5x-6 = 0
=> (x+1)(x-6) = 0
=> x = -1 or x = 6.
Thus -1 and 6 are roots of the equation.

Hence, 1, 2, 3, 4, 6 and -1 are roots of the equation.
Therefore, E is the answer.

Answer: E
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by swerve » Wed Apr 03, 2019 7:36 am
We have to consider several cases

1) \(x^2-5x+5=1\). In this case, no matter what the result of the power of this expression, the result will be \(1
x=4, x=1\).

2) \(x^2-5x-6=0\). In this case, no matter what is the value of \(x^2-5x+5\), the result will be \(1
x=6, x=-1\)

And it seemed that's all, but...

3) If \(x^2−5x+5=-1\) and \(x^2-5x-6=\) even integer the result will be \(1\)

\(x^2−5x+5=-1\) has the values \(x=3, x=2\) Both of these roots give us even result in \(x^2-5x-6\), so both roots are valid

\(X\) can have \(6\) roots: \(4, 1, 6, -1, 3, 2\).

Therefore, __E__ is the correct answer.
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