Answer to Question #142694 in Algebra for mkami

Question #142694
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).
1
Expert's answer
2020-11-09T20:12:36-0500

Let f(x)=ax2+bx+c,a0.f(x)=ax^2+bx+c, a\not=0. Then the discriminant if f(x)f(x) is


D=b24acD=b^2-4ac

The equation f(x)=6x18f(x)=6x-18 has exactly one root


ax2+bx+c=6x18ax^2+bx+c=6x-18

ax2+(b6)x+(c+18)=0ax^2+(b-6)x+(c+18)=0

(b6)24a(c+18)=0(b-6)^2-4a(c+18)=0

The equation f(x)=93xf(x)=9-3x has exactly one root


ax2+bx+c=93xax^2+bx+c=9-3x

ax2+(b+3)x+(c9)=0ax^2+(b+3)x+(c-9)=0

(b+3)24a(c9)=0(b+3)^2-4a(c-9)=0


We have the system


b212b+364ac72a=0b^2-12b+36-4ac-72a=0b2+6b+94ac+36a=0b^2+6b+9-4ac+36a=0

b24ac=12b+72a36b^2-4ac=12b+72a-36b24ac=6b36a9b^2-4ac=-6b-36a-9

12b+72a36=6b36a912b+72a-36=-6b-36a-9

b=6a+1.5b=-6a+1.5b24ac=36a936a9b^2-4ac=36a-9-36a-9

D=18D=-18



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



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