What is the sum of all numbers $a$ for which the graph of $y=x^2+a$ and the graph of $y=ax$ intersect one time?
for this to be true, x^2+a and ax are equal at only one point. that means that x^2-ax+a's discriminant is 0, which means a^2-4a is 0. that means a is 0 or 4, and the sum is 4.
HOPE THIS HELPED!
y = x^2 + a is a parabola
y = ax is a line
If the line intersects the parabola, it will be tangent to the parabola
The slope of a line tangent to the parabola at any point is 2x
So...the slope of the line = a = 2x
So....subbing in for a in both equations, we have that
y = x^2 + 2x
y = 2x (x) = 2x^2
Setting these equal we have that
x^2 + 2x = 2 x^2 rearranging, we have
x^2 - 2x = 0 factor
x( x - 2) = 0 setting each factor to 0 and solving for x produces
x = 0 or x = 2
So....a = 2(0) = 0 or a = 2(2) = 4
And the sum of these values for a is 4