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Question
Examine the nature of the roots of the quadratic (b−x)2−4(a−x)(c−x)=0 where a, b, c are real.
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Solution
The given equation can be written as 3x2−[4(a+c)−2b]x(b2−4ac)=0
Δ=[4(a+c)−2b]2+12(b2−4ac)
=4[2(a+c)−b]2+(3b2−12ac)
=4[2(a2+c2)+2ac+b2−4b(a+c)+3b2−12ac]
=16[a2+b2+c2−ab−bc−ca]
=8[(a−b)2+(b−c)2+(c−a)2]=+ive
∴ Roots are real.
Δ=[4(a+c)−2b]2+12(b2−4ac)
=4[2(a+c)−b]2+(3b2−12ac)
=4[2(a2+c2)+2ac+b2−4b(a+c)+3b2−12ac]
=16[a2+b2+c2−ab−bc−ca]
=8[(a−b)2+(b−c)2+(c−a)2]=+ive
∴ Roots are real.
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