quadratic equation

**may** · June 5th 2010, 05:27 PM

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

**Prove It** · June 5th 2010, 05:29 PM

Originally Posted by may

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

Is this $\frac{(2x - 5)(x + 1)}{x} - 1$ or $\frac{(2x - 5)(x + 1)}{x - 1}$ ?

In either case, your statement is false, as you can't divide by $0$ .

So $x \neq 0$ in the first and $x \neq 1$ in the second.

**may** · June 5th 2010, 05:34 PM

Originally Posted by Prove It

Is this $\frac{(2x - 5)(x + 1)}{x} - 1$ or $\frac{(2x - 5)(x + 1)}{x - 1}$ ?

In either case, your statement is false, as you can't divide by $0$ .

So $x \neq 0$ in the first and $x \neq 1$ in the second.

is the second one $\frac{(2x - 5)(x + 1)}{x - 1}$

**Prove It** · June 5th 2010, 05:46 PM

Originally Posted by may

is the second one $\frac{(2x - 5)(x + 1)}{x - 1}$

Your statement is still false, as $x$ can not take on the value of $1$ .

**skeeter** · June 5th 2010, 05:49 PM

Originally Posted by Prove It

Your statement is still false, as $x$ can not take on the value of $1$ .

I think the problem is referring to the range of the expression rather than the domain.

**Bacterius** · June 5th 2010, 06:21 PM

Originally Posted by may

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

I've never tried this kind of question yet, correct me if I'm wrong.

Let $f(x) = \frac{(2x - 5)(x + 1)}{x - 1}$ . If we can establish a full bijection between f $(x)$ and $x$ , then $f(x)$ must take all real values. Let's see, say $f(x) = y$ :

$y = \frac{(2x - 5)(x + 1)}{x - 1}$
$x = \frac{1}{4} \left ( - \sqrt{y^2 - 2y + 49} + y + 3 \right )$

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

$y^2 - 2y + 49$

Discriminant is $\Delta = 2^2 - 4 \times 49 = -192 < 0$ . Since the quadratic coefficient, $a$ , is positive, this quadratic yields only positive values.

Therefore the equation :

$x = \frac{1}{4} \left ( - \sqrt{y^2 - 2y + 49} + y + 3 \right )$

Does accept all values of $y$ . And therefore, in :

$y = \frac{(2x - 5)(x + 1)}{x - 1}$

$y$ does take all real values. $f(x)$ does take all real values for $x \in \mathbb{R}$ . QED.

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