Graveh what is the value of x
Algebra
Level
2
Find the real value of x in the problem (x^2-4x-4)^(x^2+5x+4) = 1.
4,1,5,1
4, -1, 5, 1
-4, -1, 5, -1
-4, -1, -5, -1
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1 solution
First, we realize that any real number raised to the 0th power will be equal to 1. So, we solve for the exponent to be 0.
We have x 2 + 5 x + 4 = 0 .
We can factor it to be ( x + 1 ) ( x + 4 ) = 0 . Quickly we realize that x = − 1 , − 4 .
We have two other cases. Either the polynomial that is being raised ( x 2 − 4 x − 4 ) is 1, or it is -1 and the exponent is a multiple of 2, causing the negative to be cancelled out.
C a s e 1 : x 2 − 4 x − 4 = 1 , x 2 − 4 x − 5 = 0 , ( x − 5 ) ( x + 1 ) = 0 , x = 5 , − 1
C a s e 2 : x 2 − 4 x − 4 = − 1 , x 2 − 4 x − 3 = 0
Using the determinant, we get that there is no rational solution, and there are no irrational solutions in the answer choices.
So, the values of x are − 4 , − 1 , 5 , − 1
However, -1 is repeated twice.