Parabola problem - Solving for an unknown 'constant'

• May 5th 2011, 09:16 AM
gundanium
Parabola problem - Solving for an unknown 'constant'
The problem that I've been trying to figure out for the last couple hours:

When the parabola y = x^2 - a(x+1) +3 intersects the x-axis at one point,
then a= 1. ____ or 2. _____

Of course you set the y = 0
then while solving it comes to a point where I have to use the quadratic formula and comes out to:

a +- sqrt{(a+6)(a-2)}
2

After that, I don't know what to do. How do I apply the statement 'intersects the x-axis at one point' to the formula?
• May 5th 2011, 09:21 AM
TheEmptySet
Quote:

Originally Posted by gundanium
The problem that I've been trying to figure out for the last couple hours:

When the parabola y = x^2 - a(x+1) +3 intersects the x-axis at one point,
then a= 1. ____ or 2. _____

Of course you set the y = 0
then while solving it comes to a point where I have to use the quadratic formula and comes out to:

a +- sqrt{(a+6)(a-2)}
2

After that, I don't know what to do. How do I apply the statement 'intersects the x-axis at one point' to the formula?

If a parabola has only 1 x intercept then its discrimiant must equal to zero.

Remember that the discrimiant is the part under the radical in the quadratic formula

$B^2-4AC$

$(-a)^2-4(1)(3-a)=0 \iff a^2+4a-12=0$
• May 5th 2011, 09:21 AM
SpringFan25
if it only intersects the x axis once, then there is only 1 solution. So the bit under the sqrt must be zero (since +0 = -0)

edit i was too slow :P
• May 5th 2011, 09:39 AM
gundanium
Ohh I see. Thank you very much
But just for a clearer understanding for future problems. Why is it that when a parabola intersects only one point on the x-axis it means its discriminant = 0?
Is this also true for horizontal axis parabolas when only hitting the y-axis once?
• May 5th 2011, 10:20 AM
Quacky
Think about how you solve a quadratic. Compare $x^2+2x-24$ (here) with $x^2+2x+1$ (here)

Try and find the zeros of both with the quadratic formula, it should become clear.

As for the horizontal axis parabolas, I can't see why it would be any different.
• May 5th 2011, 10:28 AM
gundanium
I should revise my question. What does the discriminant mean for a parabola? Is it the focus coordinate? Is it where the parabola intersects with the axis?