Range of an expression related to coefficients of

Reuben Brennan

Reuben Brennan

Answered question


Range of an expression related to coefficients of a quadratic equation
Let ax2+bx+c=0 be a quadratic equation such that both the roots lie in [0,1]. What is the range of the expression (ab)(2ab)a(ab+c)?

Answer & Explanation



Beginner2022-04-02Added 17 answers

where s=ba is the root sum and p=ca is the root product.
s[0,2] and for a fixed s, p attains a maximum of s24 (when the roots are equal) and a minimum of 0 (if s1) or s1 (if s1). Accordingly, to minimise the expression we must p=s24, whereupon it becomes
and clearly this is minimised at s=0, i.e. the lower bound on range is 2. To maximise the expression we consider two cases:
- If s[0,1] we set p=0 and the expression simplifies to 2+s, whose maximum is 3.
- If s[1,2] we set p=s1 and the expression simplifies to (1+s)(2+s)2s=1s+32+s2, which is decreasing on [1,2] and increasing otherwise. Thus we check the value at s=2, which is also 3, and conclude that the maximum is still 3.
Hence the expression range is [2, 3].

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