Answer to Question #142694 in Algebra for mkami

Question

Let us consider a quadratic polynomial f(x) such that equation f(x) = 6x - 18 has exactly one
root, and equation f(x) = 9 - 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 $f(x)=6x-18$ has exactly one root

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

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

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

The equation $f(x)=9-3x$ has exactly one root

$$ax^2+bx+c=9-3x$$

$$ax^2+(b+3)x+(c-9)=0$$

$$(b+3)^2-4a(c-9)=0$$

We have the system

$$b^2-12b+36-4ac-72a=0$$ $$b^2+6b+9-4ac+36a=0$$

$$b^2-4ac=12b+72a-36$$ $$b^2-4ac=-6b-36a-9$$

$$12b+72a-36=-6b-36a-9$$

$$b=-6a+1.5$$ $$b^2-4ac=36a-9-36a-9$$

$$D=-18$$

The value of the discriminant of $f(x)$ is $-18.$