Question 8.

The quadratic equation of the form ∣$x^2$−4x−13∣=r has its minimum value at x = -b/2a, 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 : ∣$x^2$−4⋅x−13∣ as per the given condition, this must-have 3 real roots.

The curve ABCDE represents the function ∣$x^2$−4⋅x−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, = $2^2$−8−13

= -17, the reflection about the x-axis is 17.

Alternatively,

∣$x^2$−4x−13∣=r

This can represented in two parts :

$x^2$−4x−13 = r if r is positive.

$x^2$−4x−13 = −r if r is negative.

Considering the first case : $x^2$−4x−13 =r

The quadratic equation becomes : $x^2$−4x−13−r = 0

The discriminant for this function is : $b^2$−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 :

$x^2$−4x−13+r = 0 the function inside the modulus is negative

The discriminant is 16 − (4⋅(r−13)) = 68−4r

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.