Question 767720
<font face="Times New Roman" size="+2">


There are no real number values of *[tex \LARGE k] such that *[tex \LARGE f(x)\ =\ x^2\ +\ 2kx\ +\ \frac{3}{4}\ -\ k].  For any quadratic function of the form *[tex \LARGE ax^2\ +\ bx\ +\ c], the *[tex \LARGE y]-intercept is *[tex \LARGE (0,\,c)]


Since the real values of *[tex \LARGE y]-coordinates of points on the *[tex \LARGE y]-axis are in the range *[tex \LARGE \left(-\infty,\,\infty\right)], and *[tex \LARGE \forall\ k\ \in\ \mathbb{R},\ \left(\frac{3}{4}\ -\ k\right)\ \in\ \left(-\infty,\,\infty\right)], there is no possible value of *[tex \LARGE k] such that *[tex \LARGE \left(\frac{3}{4}\ -\ k\right)] does not map to some available *[tex \LARGE y].


By the way, be careful with your terminology.  *[tex \LARGE x^2\ +\ 2kx\ +\ \frac{3}{4}\ -\ k] is NOT an equation.  Equations have equals signs in them; hence the name.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>