一元二次函数、方程和不等式

作差法

作商法

作差

变形

判断差的符号

作出结论

对称性：a=b<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Leftrightarrow"><span></span><span></span></span>b=a

传递性：a=b,b=c<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Rightarrow"><span></span><span></span></span>a=c

可加性：a=b<span class="equation-text" data-index="0" data-equation="\Leftrightarrow" contenteditable="false"><span></span><span></span></span>a<span class="equation-text" data-index="1" data-equation="\pm" contenteditable="false"><span></span><span></span></span>c=b<span class="equation-text" contenteditable="false" data-index="2" data-equation="\pm"><span></span><span></span></span>c

可乘性：a=b<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Rightarrow"><span></span><span></span></span>ac=bc

可除性：a=b,c≠0⇒<span class="equation-text" data-index="0" data-equation="\frac{a}{c}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{b}{c}"><span></span><span></span></span>

对称性：a&gt;b<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Leftrightarrow"><span></span><span></span></span>b&lt;a

传递性：a&gt;b,b&gt;c<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Rightarrow"><span></span><span></span></span>a&gt;c

可加性：a&gt;b<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Rightarrow"><span></span><span></span></span>a+c&gt;b+c

可乘性：a&gt;b,c&gt;0<span class="equation-text" data-index="0" data-equation="\Rightarrow" contenteditable="false"><span></span><span></span></span>ac&gt;bc;a&gt;b,c&lt;0<span class="equation-text" contenteditable="false" data-index="1" data-equation="\Rightarrow"><span></span><span></span></span>ac&lt;bc

同向可加性：a&gt;b,c&gt;d<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Rightarrow"><span></span><span></span></span>a+c&gt;b+d

同向可乘性：a&gt;b&gt;0,c&gt;d&gt;0<span class="equation-text" contenteditable="false" data-index="0" data-equation="\Rightarrow"><span></span><span></span></span>ac&gt;bd

可乘方性：a&gt;b&gt;0<span class="equation-text" data-index="0" data-equation="\Rightarrow" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="a^n" contenteditable="false"><span></span><span></span></span>&gt;<span class="equation-text" contenteditable="false" data-index="2" data-equation="b^n"><span></span><span></span></span>(nN,n&gt;1)

若a,b<span class="equation-text" contenteditable="false" data-index="0" data-equation="\in"><span></span><span></span></span>R,则<span class="equation-text" data-index="1" data-equation="a^2" contenteditable="false"><span></span><span></span></span>+<span class="equation-text" data-index="2" data-equation="b^2" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="3" data-equation="\geq" contenteditable="false"><span></span><span></span></span>2ab,当且仅当a=b时等号成立

若a&gt;0,b&gt;0,即<span class="equation-text" data-index="0" data-equation="\frac{a+b}{2}" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\geq" contenteditable="false"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="2" data-equation="\sqrt{ab}"><span></span><span></span></span>,当且仅当a=b时等号成立

如果积xy等于定值P，那么当x=y时，和x+y有最小值2<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sqrt{P}"><span></span><span></span></span>

如果积xy等于定值S，那么当x=y时，积xy有最大值<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{4}S^2"><span></span><span></span></span>

积定和最小，和定积最大

应用基本不等式求最值需满足”一正二定三相等“

构造定值条件的通常技巧

我们把只含有一个未知数，并且未知数的最高次数是2的不等式，例如a<span class="equation-text" data-index="0" data-equation="x^2" contenteditable="false"><span></span><span></span></span>+b<span class="equation-text" data-index="1" data-equation="x" contenteditable="false"><span></span><span></span></span>+c<span class="equation-text" contenteditable="false" data-index="2" data-equation="\geq"><span></span><span></span></span>0（其中a,b,c均为常数、a≠0)，叫做一元二次不等式

一般地，对于二次函数y=a<span class="equation-text" data-index="0" data-equation="x^2" contenteditable="false"><span></span><span></span></span>+b<span class="equation-text" data-index="1" data-equation="x" contenteditable="false"><span></span><span></span></span>+c,我们把使a<span class="equation-text" data-index="2" data-equation="x^2" contenteditable="false"><span></span><span></span></span>+b<span class="equation-text" data-index="3" data-equation="x" contenteditable="false"><span></span><span></span></span>+c=0的实数<span class="equation-text" data-index="4" data-equation="x" contenteditable="false"><span></span><span></span></span>叫做二次函数a<span class="equation-text" data-index="5" data-equation="x^2" contenteditable="false"><span></span><span></span></span>+b<span class="equation-text" contenteditable="false" data-index="6" data-equation="x"><span></span><span></span></span>+c的零点

将不等式化为右边为零，左边为二次项系数大于零的不等式a<span class="equation-text" data-index="0" data-equation="x^2" contenteditable="false"><span></span><span></span></span>+b<span class="equation-text" data-index="1" data-equation="x" contenteditable="false"><span></span><span></span></span>+c&gt;0(a&gt;0)或a<span class="equation-text" data-index="2" data-equation="x^2" contenteditable="false"><span></span><span></span></span>+b<span class="equation-text" contenteditable="false" data-index="3" data-equation="x"><span></span><span></span></span>+c&lt;0(a&gt;0)

求出相应的一元二次方程的根

利用二次函数的图像与轴的交点确定一元二次不等式的解集