Show that if x is real ,the expression [(2x-5)(x+1)]/x-1 can take all the real values.

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- Jun 5th 2010, 05:27 PMmayquadratic equation
Show that if x is real ,the expression [(2x-5)(x+1)]/x-1 can take all the real values.

- Jun 5th 2010, 05:29 PMProve It
- Jun 5th 2010, 05:34 PMmay
- Jun 5th 2010, 05:46 PMProve It
- Jun 5th 2010, 05:49 PMskeeter
- Jun 5th 2010, 06:21 PMBacterius
*I've never tried this kind of question yet, correct me if I'm wrong.*

Let . If we can establish a full bijection between f and , then must take all real values. Let's see, say :

The square root is bothersome, as we need to show that the value under is (hopefully) always positive. So :

Discriminant is . Since the quadratic coefficient, , is positive, this quadratic yields only positive values.

Therefore the equation :

Does accept all values of . And therefore, in :

does take all real values. does take all real values for . QED.