二次方程的根

爱德思国际普通中等教育证书进阶纯数学· S2· 25 分钟阅读

1. 通过因式分解法求解二次方程★★☆☆☆⏱ 8 min

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📘 定义

二次方程的根

满足方程 $a x^{2} + b x + c = 0$ 的 $x$ 取值，也被称为二次方程的解或二次函数图像的x轴截距。

例:

$x = 2$ and $x = - 3$ are roots of $x^{2} + x - 6 = 0$

对于可因式分解的二次方程，因式分解法是最快的求解方法：你可以将二次式改写为两个一次因式的乘积，再令每个因式等于零即可求出根。该方法仅适用于可在整数范围内因式分解的二次式。

📐 例题

用因式分解法求解方程 $2 x^{2} + 7 x - 4 = 0$ 。

1
找到两个整数，使其乘积为 $2 \times - 4 = - 8$ 且和为7：即8和-1。
2
拆分中间项并分组因式分解： $2 x^{2} + 8 x - x - 4 = 2 x (x + 4) - 1 (x + 4) = (2 x - 1) (x + 4)$
3
令每个因式等于零： $2 x - 1 = 0 ⟹ x = \frac{1}{2}$ ； $x + 4 = 0 ⟹ x = - 4$
4
核验两个解都满足原方程，确认没有计算错误。

Exam tip:

考试时如果时间充裕，务必将因式分解得到的根代回原方程进行核验，避免粗心错误。

2. 通过配方法求解二次方程★★★☆☆⏱ 10 min

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配方法将二次式改写为顶点形式 $a (x + p)^{2} + q = 0$ ，再通过移项单独分离出 $x$ 即可求解根。该方法适用于所有存在实根的二次方程，也是二次求根公式的推导基础。

📐 例题

用配方法求解方程 $x^{2} - 6 x + 2 = 0$ ，结果保留精确形式。

1
分离常数项： $x^{2} - 6 x = - 2$
2
对含 $x$ 的项配方：-6的一半是-3，平方后为9，在等式两边同时加9： $x^{2} - 6 x + 9 = - 2 + 9 = 7$
3
改写为平方项形式： $(x - 3)^{2} = 7$
4
对等式两边同时开平方（保留 $\pm$ ）： $x - 3 = \pm 7$
5
整理得到 $x$ ： $x = 3 + 7$ 或 $x = 3 - 7$

3. 使用二次求根公式求解二次方程★★☆☆☆⏱ 7 min

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📘 定义

二次求根公式

x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}

用于求解任意满足 $a \neq = 0$ 的二次方程 $a x^{2} + b x + c = 0$ 的通用公式。该公式需要你在考试前熟记，考场上不会提供公式表。

该方法适用于所有二次方程，包括无法因式分解、或要求小数结果的二次方程。代入数值时请特别注意 $b$ 、 $c$ 为负数的情况，以及判别式部分的计算。

📐 例题

使用二次求根公式求解方程 $3 x^{2} - 5 x - 1 = 0$ ，结果保留两位小数。

1
从标准形式方程中识别参数： $a = 3$ ， $b = - 5$ ， $c = - 1$ 。
2
代入公式： $x = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( 3 ) ( - 1 )}{2 ( 3 )} = \frac{5 \pm 25 + 12}{6}$
3
化简判别式： $37 \approx 6.0828$
4
计算两个根： $x = \frac{5 + 6.0828}{6} \approx 1.85$ ， $x = \frac{5 - 6.0828}{6} \approx - 0.18$

Exam tip:

代入负的 $b$ 值时，建议在负号外添加括号避免符号错误，例如写 $- (- 5)$ 而不是直接写 $- - 5$ 。

4. 使用判别式对根进行分类★★☆☆☆⏱ 6 min

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📘 定义

判别式

\Delta = b^2 -4ac

二次求根公式中位于平方根下的表达式，可用于在不完全求解的情况下判断二次方程根的数量和类型。

If $Δ > 0$ : 2 distinct, unequal real roots
If $Δ = 0$ : 1 repeated, equal real root
If $Δ < 0$ : no real roots (roots are non-real complex)

📐 例题

判定方程 $2 x^{2} - 4 x + 3 = 0$ 的根的性质。

1
识别系数 $a = 2$ ， $b = - 4$ ， $c = 3$ 。
2
计算判别式： $Δ = (- 4)^{2} - 4 (2) (3) = 16 - 24 = - 8$
3
由于 $Δ = - 8 < 0$ ，该二次方程没有实根。

✓ 快速检测

What is the nature of roots of $x^{2} - 8 x + 16 = 0$ ?
- Two unequal real roots
- One equal real root
- No real roots
显示答案
1 —
Discriminant = $64 - 64 = 0$ , so there is one repeated real root.

5. 常见陷阱

错误做法:

Forgetting to include the $\pm$ symbol when taking square roots during completing the square or using the quadratic formula.

原因:

This causes you to miss one of the two roots, losing half the available marks for the question.

正确做法:

Always write the $\pm$ sign immediately after the square root symbol when solving for $x$ .

错误做法:

Incorrectly identifying $a$ , $b$ , $c$ values when the quadratic is not in standard form $a x^{2} + b x + c = 0$ .

原因:

Substituting wrong values into the quadratic formula or discriminant gives completely incorrect results.

正确做法:

Rearrange the quadratic into standard form, with all terms on one side equal to zero, before identifying $a$ , $b$ , $c$ .

错误做法:

Making sign errors when substituting negative values of $b$ into the quadratic formula, e.g., writing $- 5$ instead of $- (- 5)$ when $b = - 5$ .

原因:

This flips the sign of the numerator, leading to wrong root values.

正确做法:

Wrap all negative values in parentheses when substituting into formulas to avoid sign errors.

错误做法:

Assuming all quadratics can be factorised, wasting time trying to factorise when the quadratic is not solvable over integers.

原因:

You lose time in the exam that could be spent on other questions.

正确做法:

If factorisation does not work after 30 seconds, switch to the quadratic formula or completing the square method.

错误做法:

Confusing discriminant conditions, e.g., stating $Δ > 0$ means equal roots.

原因:

This leads to incorrect classification of root type, losing all marks for that part of the question.

正确做法:

Memorise the three discriminant cases, and double-check your calculation of $Δ$ before classifying roots.

6. 速查表

Method	Use Case	Key Steps / Rules
Factorisation	Quadratic factors easily over integers	Rewrite as product of linear factors Set each factor = 0 Solve for x
Completing the Square	Exact root answers, vertex form required	Isolate constant term Add square of half coefficient of x to both sides Rearrange to solve for x, include $\pm$
Quadratic Formula	All quadratics, decimal answers required	Recall formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ Substitute a,b,c correctly Simplify to required precision
Discriminant	Classify root type without solving	$Δ = b^{2} - 4 a c$ $Δ > 0$ : 2 unequal real roots $Δ = 0$ : 1 equal real root $Δ < 0$ : No real roots

深入阅读

syllabus爱德思4PM1考试大纲2016年首次开课批次的官方教学大纲

下一步

Now that you have mastered solving quadratic equations and classifying their roots, you are ready to move on to more advanced quadratic function topics in the Edexcel IGCSE Further Pure Mathematics syllabus. The next core sub-topic covers symmetric functions of roots, where you will learn to use the sum and product of roots without solving the full quadratic equation. You can also practice applying these root-solving skills to coordinate geometry problems involving quadratic graphs, which appear frequently in exam papers. Make sure to practice past paper questions to reinforce your understanding of all three solution methods and discriminant classification, as these skills are foundational for almost all other algebra topics in the course.