二次函数f(x)=ax2+bx+c(a≠0),若a>0,△=b2-4ac≤0,则f(x)≥0;若a<0,△=b2-4ac≤0,则f(x)≤0. 二次方程ax2+bx+c=0(a≠0)有实根,则△=b2-4ac≥0. 以上性质,我们可以用来证明不等式. 例1 已知a,b∈R,且b>0.求证:a2+b2>3a-2ab-3. 证明:被证不等式可变形为 The quadratic function f(x)=ax2+bx+c(a≠0), if a>0, △=b2-4ac≤0, then f(x)≥0; if a<0, △=b2-4ac ≤ 0, then f (x) ≤ 0. The quadratic equation ax2 + bx + c = 0 (a ≠ 0) has a real root, then △ = b2-4ac ≥ 0. The above properties, we can use to prove inequality. 1 Known a, b∈R, and b>0. Proof: a2+b2>3a-2ab-3. Proof: The proven inequality can be transformed into