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  t = ±√3

Step-by-step explanation:

The first equation tells us ...

  x^2 +tx +2 = (x -h)(x -k)

 x^2 +tx +2 = x^2 -(h+k)x +hk

Comparing coefficients, we have the equations ...

  t=-(h+k)

  2 = hk

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The second equation tells us ...

  2x^2 +mx +2m = 2(x -h/k)(x -k/h)

  2x^2 +mx +2m = 2x^2 -2(h/k +k/h)x +2

Comparing coefficients, we have the equations ...

  m = -2(h/k +k/h)

  2m = 2

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Dividing the last equation by 2 gives ...

  m = 1

Using that in the third equation, we have ...

  1 = -2(h/k +k/h) = -2(h^2 +k^2)/hk

Using hk = 2, this gives us ...

  h^2 +k^2 = -1

From the first equation, we know ...

  t^2 = (-(h+k))^2 = h^2 +2hk +k^2 = (h^2 +k^2) +2(hk)

Substituting the values for these terms, we have ...

  t^2 = -1 + 2(2) = 3

  t = ±√3

Possible values of t are √3 and -√3.

a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. because no real number satisfies this equation, i is called an imaginary number. has no real solution, since the square of a real number cannot be negative.

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step-by-step explanation: