The real quadratics ax2 + 2bx + c and Ax2 + 2Bx + C are non-negative for all real x. Show that aAx2 + 2bBx + cC is also non-negative for all real x.
Solution
ax2 + 2bx + c is non-negative for all real x iff a ≥ 0 and b2 -ac ≥ 0. So a, A ≥ 0 and hence aA ≥ 0, and b2 ≥ ac and B2 ≥ AC and hence (bB)2 ≥ (aA)(cC), so aAx2 + 2bBx + cC is non-negative for all real x.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03