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如何因式分解二次多项式（二次方程）

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本文将教你如何因式分解二次多项式。一个多项式含有一个变量（x），x有特定的次数，多项式还有各种其他的变量和常数。要因式分解一个二次多项式成多个多项式因子相乘的形式，你的数学水平得达到代数I以上，否则不太容易理解本方法的原理。本文中都用到的标准形式的二次多项式：ax ² + bx + c = 0

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以次数高低排列，如果有最大公因数则提出来：6 + 6x ² + 13x，6x ² + 13x + 6
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用FOIL（首项相乘、外项相乘、内向相乘、次项相乘，这是展开多项式相乘的一种步骤方法）分解，并合并同类项：(2x + 3)(3x + 2)，6x ² + 4x + 9x + 6，6x ² + 13x + 6。

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方法 1

方法 1 的 6:

试错法

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若你的多项式十分简单，可以自己来发现因数。注意：用这个方法，可能不能因式分解更复杂的三项式了。例子: 3x ² + 2x - 8

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1 和 3，c = -8 因数: 2 和 4 和 1 和 8
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以下是方法：如果ax ² + bx + c 则 (x + h)(x + k)，如果 ax ² - bx - c 或 ax ² + bx - c 则 (x - h)(x + k)。如果 ax ² - bx + c 则 (x - h)(x - k)。本例子中是 3x ² + 2x - 8 ，因此 (x - h)(x + k)是答案的形式，然后试试： (3x + 8)(x - 1)
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(3x + 8)(x - 1)，3x ² - 3x + 8x - 8，3x ² + 5x - 8 ≠ 3x ² + 2x - 8
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7
如果必要，则换掉因数。 本例中我们试试2和4这对： (3x + 2)(x - 4)
- c 现在是-8。
- 但是外项和内项积分别是-12x 和 2x，合并不成+2x。
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8
如果必要的话就调转顺序。 我们试试把2、4换个位置。 (3x + 4)(x - 2)
- c 还是对的。
- 外项积和内项积是-6x 和 4x，则这两个数的和同2x正好符号相反
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9
然后再确认一下符号正负。 顺序是没错的，现在把符号倒过来： (3x - 4)(x + 2)
- c 还是对的。
- 外项积和内项积现在6x 和 -4x。加起来等于2x ，这次就对了。
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方法 2

方法 2 的 6:

分解法

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不喜欢猜的方法，可以试试这个。

例子: 6x ² + 13x + 6

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ax ² + kx + hx + c，6x ² + 4x + 9x + 6
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整理一下方程，使得可以提出最大公因式（(3x+2)），然后合并同类项，得到因式分解结果。6x ² + 4x + 9x + 6，2x(3x + 2) + 3(3x + 2)，(2x + 3)(3x + 2)

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方法 3

方法 3 的 6:

三重方法

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本方法很像分解法，不过更简单例子: 8x ² + 10x + 2

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8•2 = 16
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2•8 = 16 8 + 2 = 10
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将两个数（ h 、 k ）代入这个方程：(ax + h)(ax + k)---------------------- a(8x + 8)(8x + 2)---------------------- 8（如图）
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a {本例中为(8x + 8)}。用 a 除以这个数，让另一项保持原样(8x + 8)(8x + 2)---------------------- 8，答案:(x + 1)(8x + 2)
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方法 4

方法 4 的 6:

两个平方之差

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27x ² - 12，3(9x ² - 4)
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一定要有两项，否则不能平均分解这个方程。√(9x ² ) = 3x ， √(4) = 2 （注意这里省去了负数根。）
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把“a”、“c”从你的等式中代入下列公式：(√(a) + √(c))(√(a) - √(c))3[(√(9x ² ) + √(4))(√(9x ² ) - √(4))]3[(3x + 2)(3x - 2)]

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方法 5

方法 5 的 6:

使用二次公式

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上述方法都不行，则用二次公式例如：x ² + 4x + 1

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将对应量代入本方程：x = -b ± √(b ² - 4ac) --------------------- 2a，x = -4 ± √(4 ² - 4•1•1) ----------------------- 2•1（如图）
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得到两个x，x= -4 ± √(16 - 4) ------------------ 2x = -4 ± √(12) -------------- 2x = -4 ± √(4•3) -------------- 2x = -4 ± 2√(3) -------------- 2x = -2 ± √(3)，x = -2 + √(3) 或 x = -2 - √(3)（如图）
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方法 6

方法 6 的 6:

用计算器

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这些步骤适合TI图形计算器，在标准考试中尤其好用。

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3
找到和x 轴相交点得到(-1, 0), (2 , 0)，x = -1, x = 2
- 如果看不到，则按下[2nd] -[TRACE]，按下 [2] 或选择“0”。移到交点之左以后按下[ENTER]，移到交点之右按下[ENTER]，移到尽量接近和x轴相交的点旁边，按下 [ENTER]，计算器就会自动算出该点的横坐标。对另一个交点也重复此步骤。
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把x值( h 和 k )代入本公式： (x - h)(x - k)，(x - (-1))(x - 2)，整理为(x + 1)(x + (-2)) 表示出两个交点来。

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利用箱型法（可视解）

本网站有解释: http://www.purplemath.com/modules/factquad3.htm

视频说明: http://www.youtube.com/watch?v=bq1Iw1w1Bgo

小提示

若用二次公式因式分解了一个多项式，其中含有根数，可能需要将x换成分数来检查该解是否正确。
如果一个项没有系数，则系数是1。x ² = 1x ²
如果有 TI-84 计算器 (可画图) ，则有一个叫做SOLVER的程序可以解二次方程，这个程序还可以解任何其他次数的多项式。
如果一个项不存在，则它的系数是0。有时把0项写出来会比较方便，比如x ² + 6 = x ² + 0x + 6
在熟悉试错法前，先把要试的因数写下来，熟练了以后再在脑子中运算。

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警告

如果你在数学课中学到了这个概念，要注意老师建议用什么方式，就尽量不要用你喜欢的别的方式。因为老师可能会让你在考试中用特定的一种方法来解，或者不让你用图画计算器来解。

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你需要准备

铅笔
纸
二次方程，或叫二次多项式
画图计算器（可选）

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