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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?

Jan 13, 2019

#1
+533
0

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!

Jan 13, 2019
#2
+111437
+1

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

Jan 13, 2019
#3
0

thank you both!

Guest Jan 13, 2019