这部分需要会什么？
- 在展开式、因式分解式和顶点式之间转换
- 使用平方差与完全平方公式
- 通过等价形式直接读取所需信息
核心概念与公式
二次式的展开式为 \(ax^2+bx+c\)，其中 \(a\ne0\)。有实数零点时可写成 \(a(x-r_1)(x-r_2)\)，便于读零点；顶点式 \(a(x-h)^2+k\) 便于读顶点。
常用恒等式为 \((u+v)^2=u^2+2uv+v^2\)、\((u-v)^2=u^2-2uv+v^2\) 和 \(u^2-v^2=(u-v)(u+v)\)。配方时，先提出二次项系数，再在括号内补上一次项系数一半的平方，同时抵消所补的量。
做题时怎样使用
- 需要零点时优先因式分解，需要最值或顶点时优先配方
- 配方前将二次项系数提出，注意括号外系数对补偿常数的影响
- 将改写后的式子展开一次，核对每一项
容易混淆的地方
\((u+v)^2\) 不能漏掉 \(2uv\)。二次式没有实零点时，不能强行分解成两个实系数一次因式。
用一个例题串起方法
知识点教学例题将 \(2x^2-12x+11\) 写成顶点式。
展开解题思路 ＋
\(2x^2-12x+11=2(x^2-6x)+11=2[(x-3)^2-9]+11=2(x-3)^2-7\)。因此顶点为 \((3,-7)\)。
本知识点题库
先独立作答，再查看答案与解析。
二次式展开・配方与等价形式挑战
Which expression is equivalent to \( (3x+7)^2-(x-6) \)?
参考答案
二次式展开・配方与等价形式进阶
Which of the following is a factor of \( −24x^2−6x \)?
参考答案
二次式展开・配方与等价形式挑战
The expression \( (7x+y)(x+7y)(3x+y) \) is equivalent to \( ax^3+bx^2y+cxy^2+dy^3 \), where a,b,c and d are constants. What is the value of c?
参考答案
二次式展开・配方与等价形式进阶
Which expression is equivalent to \( 9x^3+72x^2y+xy^2+8y^3 \)?
参考答案
二次式展开・配方与等价形式挑战
Which of the following expressions is(are) a factor of \( 5x^2+22x−48 \)?
Ⅰ. x−6
Ⅱ. 5x−8
参考答案
二次式展开・配方与等价形式进阶
Which expression is equivalent to \( 7x^3+4x-5-9x^2+6x+3 \) ?
参考答案
二次式展开・配方与等价形式挑战
\( 51x^{14}+bx^7+38 \)
The given expression, where b is a constant, is equivalent to \( (3x^7+q)(rx^7+2) \), where q and r are constants. What is the value of b?
The given expression, where b is a constant, is equivalent to \( (3x^7+q)(rx^7+2) \), where q and r are constants. What is the value of b?
参考答案
二次式展开・配方与等价形式挑战
\( (4x+10)(2x+1)−(2x−14) \)
For all values of x, the given expression is equivalent to which of the following?
I. \( 8x^2+22x+24 \)
II. \( 8x^2+24x+10-2x-14 \)
For all values of x, the given expression is equivalent to which of the following?
I. \( 8x^2+22x+24 \)
II. \( 8x^2+24x+10-2x-14 \)
参考答案
二次式展开・配方与等价形式挑战
\( 57x^{10}+bx^5+26 \)
The given expression, where b is a constant, is equivalent to \( (3x^5+q)(rx^5+2) \) , where q and r are constants. What is the value of b?
The given expression, where b is a constant, is equivalent to \( (3x^5+q)(rx^5+2) \) , where q and r are constants. What is the value of b?
参考答案
二次式展开・配方与等价形式基础
Which expression is equivalent to \( 20x^2 \)?
参考答案
二次式展开・配方与等价形式进阶
The expression \( (3x+4)(x+6) \) can be written in the form \( ax^2+bx+c \), where a, b, and c are constants. What is the value of \( a+b+c \)?
参考答案
二次式展开・配方与等价形式基础
Which expression is equivalent to \( 143x^2 \)?
参考答案
