Question

What are the possible value of x any software can't get all values but you can

Answer 1

Answer:

  x ∈ {2, 3, 4, 5, 6, 7}

Step-by-step explanation:

The equation you have written only has two real solutions. Perhaps you intend ...

  $(x^2-7x+11)^{x^2-13x+42}=1$

The base quadratic equation factors as ...

  (x -5)(x -2) +1

so will have the value 1 for x=2 and x=5. The exponent quadratic can have any value and the intended equation will be satisfied.

__

The exponent quadratic factors as ...

  (x -6)(x -7)

so will be zero for x=6 and x=7. The base quadratic can have any non-zero value and the intended equation will be satisfied.

So, the four solutions found by a typical solver or graphing program are ...

  x ∈ {2, 5, 6, 7}

___

In general, for values of x between 2.382 and 4.618, the base quadratic will be negative and the left side of the equation will be complex. However, there are 10 values of x where the imaginary part is zero, and two of these where the real part is 1. These values of x are x=3 and x=4, where the equation is (-1)^6 = 1 and (-1)^12=1.

In summary, the solutions are ...

  x ∈ {2, 3, 4, 5, 6, 7}

Answer 2

Answer:

x ∈ {2, 3, 4, 5, 6, 7}

Step-by-step explanation:

The equation you have written only has two real solutions. Perhaps you intend ...

 

The base quadratic equation factors as ...

 (x -5)(x -2) +1

so will have the value 1 for x=2 and x=5. The exponent quadratic can have any value and the intended equation will be satisfied.

__

The exponent quadratic factors as ...

 (x -6)(x -7)

so will be zero for x=6 and x=7. The base quadratic can have any non-zero value and the intended equation will be satisfied.

So, the four solutions found by a typical solver or graphing program are ...

 x ∈ {2, 5, 6, 7}

___

In general, for values of x between 2.382 and 4.618, the base quadratic will be negative and the left side of the equation will be complex. However, there are 10 values of x where the imaginary part is zero, and two of these where the real part is 1. These values of x are x=3 and x=4, where the equation is (-1)^6 = 1 and (-1)^12=1.

In summary, the solutions are ...

 x ∈ {2, 3, 4, 5, 6, 7}

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