The quadratic equation of the form x2−4x−13=r has its minimum value at x=2a−b 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=0r=17 for r=17 we can have exactly 3 roots.
