For what values of p does the vertex of the parabola y=x^2 +2px+13 lie at a distance of 5 units from the origin

Dear student

 

it is not clear that distance should be towarda x axis or y axis

it is clear that it is upward parabola

 

now

 

its b/2a equals to 5

 

2p/2 = 5

 

p = 5

Find the vertex ( -p, 13-p^2 ) . 

Now using distance formula from origin , equate the distance as 5 .( Distance can be 5 in any direction  so it's totally wrong to assume it's along X or Y axis.  

 

p^2 +(13-p^2)^2 = 25 

Solve to get p = 3,-3,4,-4