Question

Let us consider a quadratic polynomial f(x) such that equation f(x)=7x-14 has exactly one root, and equation f(x)=6-3x has exactly one root. Find the minimum value of a discriminant of f(x).

Answer 1

Let $f(x)=ax^2+bx+c, a\not=0.$ Then the discriminant if $f(x)$ is

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

The equation $f(x)=7x-14$ has exactly one root

$$ax^2+bx+c=7x-14$$ $$ax^2+(b-7)x+(c+14)=0$$ $$(b-7)^2-4a(c+14)=0$$

The equation $f(x)=6-3x$ has exactly one root

$$ax^2+bx+c=6-3x$$ $$ax^2+(b+3)x+(c-6)=0$$ $$(b+3)^2-4a(c-6)=0$$

We have the system

$$b^2-14b+49-4ac-56a=0$$ $$b^2+6b+9-4ac+24a=0$$ $$b^2-4ac=14b+56a-49$$ $$b^2-4ac=-6b-24a-9$$ $$14b+56a-49=-6b-24a-9$$ $$b=-4a+2$$ $$b^2-4ac=24a-12-24a-9$$ $$D=-21$$

The value of the discriminant of $f(x)$ is $-21.$