Question

Let us consider a quadratic polynomial f(x) such that equation f(x) = 5x - 15 has exactly one root, and equation f(x) = 6x - 18 has exactly one root. Find the maximum value of a discriminant of f(x).

Answer 1

Let $f(x)=ax^2+bx+c$ be a quadratic polynomial.

The equation $f(x)=5x-15$ has exactly one root

$$ax^2+bx+c=5x-15$$

$$ax^2+(b-5)x+(c+15)=0$$

$$D_1=(b-5)^2-4a(c+15)=0$$

The equation $f(x)=6x-18$ has exactly one root

$$ax^2+bx+c=6x-18$$

$$ax^2+(b-6)x+(c+18)=0$$

$$D_2=(b-6)^2-4a(c+18)=0$$

$$(b-5)^2-(b-6)^2-4a(c+15-c-18)=0$$

$$a=-\dfrac{1}{12}(2b-11)$$

$$b^2-10b+25-4ac+5(2b-11)=0$$

$$4ac=b^2-30$$

$$D_0=b^2-4ac=b^2-(b^2-30)=30$$

The maximum value of a discriminant of f(x) is 30.