The quadratic equation of the form
|x2−4x−13|=r has its minimum value at
x=−a and hence does not vary irrespective of the value of x.
Hence at x = 2 the quadratic equation has its minimum.
Considering the quadratic part :
|x2−4x−13| as per the given condition, this must have 3 real roots.
The curve A B C D E represents the function
|x2−4x−13| because of the modulus function, the representation of the quadratic equation becomes :
ABC'DE.
There must exist a value, r such that there must exactly be 3 roots for the function. If r = 0 there will only be 2 roots, similarly for other values there will either be 2 or 4 roots unless at the point C'.
The point C' is a reflection of C about the x-axis. r is the y coordinate of the point C' :
The point C which is the value of the function at
x=2 =22−8−13=−17 the reflection about the x-axis is 17.
Alternatively
|x2−4x−13|=r This can represented in two parts:
x2−4x−13=r if
r is positive.
x2−4x−13=−r if
r is negative.
Considering the firs case :
x2−4x−13=r The quadratic equation becomes :
x2−4x−13−r=0 The discriminant for this function is :
b2−4ac=16−(4(−13−r))=68+4r Since r is positive the discriminant is always greater than 0 this must have two distinct roots.
For the second case :
x2−4x−13+r=0 the function inside the modulus is negative
The discriminant is
16−(4(r−13))=68−4r In order to have a total of 3 roots, the discriminant must be equal to zero for this quadratic equation to have a total of 3 roots.
Hence,
68−4r=0 r=17 for
r=17 we can have exactly 3 roots.