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Find all numbers a for which the graph of \(y=x^2+a\) and the graph of \(y=ax\) intersect. Express your answer in interval notation.

I tried putting in (4, inf) but it was incorrect on AoPS. Can someone help?

 Jul 27, 2019
 #1
avatar+19749 
+2

How about [4. inf) ?         and    (-inf, 0]  ?

 Jul 27, 2019
edited by ElectricPavlov  Jul 27, 2019
edited by ElectricPavlov  Jul 27, 2019
 #2
avatar+104704 
+3

y = x^2 + a

y = ax

 

Set the y's equal

 

x^2 + a  =  ax          rearrange as

 

x^2 - ax + a = 0 

 

For this to have real solutions.......the discriminant must  be ≥  0

 

So

 

a^2  -  4a ≥  0

 

a ( a - 4) ≥  0

 

Setting each factor to 0  and solving for a  produces the following solutions  a = 0   and a = 4

 

So  we have the following possible intervals that  will produce solutions

 

(-inf, 0 ]  or  ( 0 , 4 )   or  [ 4, inf )

 

If a  is in the first interval, then a (a - 4) ≥ 0  so this interval produces a solution

If  a is in the second interval, then a(a - 4)  < 0.....so  no solutions are found here

If a  is in the third interval, then a (a - 4) ≥ 0 .....so this interval produces a solution

 

So....the solution intervals are  

 

(-inf, 0 ]  U [ 4, inf )...as EP found  !!!

 

 

cool cool cool

 Jul 27, 2019

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