<p>I was studying for the Math SAT II, when I ran into some difficulty into this particular problem. Any help is very much appreciated.</p>
<h2>Thank you!</h2>
<p>4x^2 - 2kx + 1 = k - 2</p>
<h2>The equation given above has two real and distinct roots. What are all the possible values of k?</h2>
<p>4x^2-2kx+1=k-2
4x^2-2kx-k+3=0
in order this equation to have two real and distinct real roots its discriminant must be bigger than zero so
4k^2-4<em>4</em>(3-k)>0 =>
4k^2+16k-48>0 =>
k^2+4k-12>0 this can be written as
(k+6)(k-2)>0 now, we got that the roots are -6 and 2, when you use a sign table you get that: K is (-inf, -6) U (2, +inf).</p>
<p>Is this a question from Barron's? I don't like Barrons...</p>
<p>argh...skopsko.........what did you do in the end....i mean what sign table?....</p>
<p>and...Barrons is a bit tougher than the rest...but obviouslly the best!</p>
<p>When people say a bit tougher, what do they mean?</p>
<p>I mean....it covers a few topics that are usually not on the actual tests. It makes u study more advanced stuff so that the actual test becmes a breeze...</p>
<p>x -inf.............Root 1................Root2..........+inf<br>
ax^2+bx+c + [ - ] + </p>
<p>Now it depends on the sign of "a" in the equation ax^2+bx+c>0 for how you are going to start from " infinity"! And then after every root you change the sign ( + goes to - ). This table is drawn as when "a" is positive coefficient. This sign table tells us that the inequality is satisfied when x<root1 and="" x="">root2 (since it is positive in these intervals).
If the equation was ax^2+bx+c<0 and "a" again positive,then again by using the same table we can find the interval where it is satisfied, namely root1<x<root2 (notice that it is negative in that interval).</root1></p>
<p>hope this helped, cause it is hard to explain it by typing :)</p>
<p>x__________ -inf.............Root 1................Root2..........+inf
ax^2+bx+c ............+.........[........... - .........].......+.........</p>
<p>I am afraid my "table" wasn't successful in the previous post:)</p>
<p>ah...thats okay.....i'll learn it from anothr book or sth.....thanx for da effort though!</p>
<p>Thank you, skopsko. I understand it quite clearly now. Your help is VERY much appreciated.</p>
<p>In reply to #3,</p>
<p>Yes, it is indeed from Barron's prep. :)</p>
<p>Let's hope it'll give me an 800 when I take the test!</p>
<p>use artpad to link us.</p>
<p><a href="http://artpad.art.com/%5B/url%5D">http://artpad.art.com/</a></p>
<p>hi everyone. I have a problem for u. i bet no one can solve this. i was doing my math test yeasterday and no one in my class could solve it exept me and a friend of mine. the problem is:
u have a parabola with equation x^2=4pY (p>0). also u have a circle whith equation X^2+Y^2=R^2 and the diameter of the circle is 3p. Now u have to proov that the comon cord of the circle and the parabola bisects seg OF. where O is the center of the circle and of the parabola and F is the... (damn it i am albanian and i dont know english terms but the coordinate of F is (b/4;0) i think you know what F is now).</p>
<p>I don't see the point of challenging the CC community.
Since you are a long-time member of CC, you would probably agree that this forum is used mostly for helping each other.</p>
<p>x^2 = 4py
x^2 + y^2 = R^2
2R = 3p</p>
<p>4py + y^2 = (3p/2)^2
4y^2 + 16py - 9p^2 = 0</p>
<p>y = (-8p +- SQRT(64p^2 + 36p^2)) / 4
y = (-8p +- 10p) / 4</p>
<p>The parabola and the circle intersect above x-axis
y>0
y = (-8p + 10p) / 4
y = p/2
p/2 is the y-coordinate of the intersection points of the circle and the parabola.
The common cord of the circle and the parabola is a segment connecting those two points.
This line intersects y-axis in B(0, p/2).</p>
<p>Focus F of a parabola x^2 = 4py is in (0, p).
OF = 2 OB.
++++++++
Need I throw some IMO question back at you (and your friend)? :D</p>
<p>ok gcf was not meant to challange u in the way u think but i was just curious to see if u could make it out, and second try to solve this exercise along with 7 pther exercises like this within a 45 min test.</p>
<p>I can't not claim I could finish this and 7 other questions of comparable difficulty in 45 min. (I am not good at sprints).
Still my question remains - what's the point in demonstrating your abilities just for the sake of showing off?</p>