Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations form the backbone of IGCSE Mathematics. From simple factorisation to the quadratic formula and graphs, this topic appears in nearly every exam paper. Understanding the structure and methods thoroughly will boost your confidence and your grade.

二次方程是 IGCSE 数学的核心内容。从简单的因式分解到求根公式和图像，这一话题几乎出现在每一份考卷中。深入理解其结构与解法，将大幅提升你的信心和成绩。

1. What is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. In its standard form, it is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是最高次数为 2 的多项式方程，即变量的最高次幂为 2。其标准形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

The coefficient a cannot be zero; if a = 0, the equation becomes linear. The constants b and c may be zero, but the x² term must always be present for the equation to be quadratic.

系数 a 不能为零；若 a = 0，方程就变成了线性方程。常数 b 和 c 可以为零，但方程中必须始终含有 x² 项，才能被称为二次方程。

Here are some examples of quadratic equations: x² + 5x + 6 = 0, 2x² − 4x = 0, and x² − 9 = 0. Each can be solved using methods we will explore below.

下面是一些二次方程的例子：x² + 5x + 6 = 0、2x² − 4x = 0 和 x² − 9 = 0。每一个都可以用我们下面将介绍的方法求解。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer roots. The goal is to write ax² + bx + c as a product of two binomials, then set each factor equal to zero.

当二次方程具有简单的整数根时，因式分解是最快捷的方法。目标是将 ax² + bx + c 写成两个二项式的乘积，然后令每个因式等于零。

Consider x² + 7x + 12 = 0. We look for two numbers that multiply to give 12 and add to give 7. Those numbers are 3 and 4, so the factorised form is (x + 3)(x + 4) = 0.

考虑 x² + 7x + 12 = 0。我们要寻找两个数，它们相乘得 12，相加得 7。这两个数是 3 和 4，因此因式分解形式为 (x + 3)(x + 4) = 0。

Now we apply the zero-product property: if (x + 3)(x + 4) = 0, then either x + 3 = 0 or x + 4 = 0. This gives the solutions x = −3 and x = −4.

现在应用零积性质：若 (x + 3)(x + 4) = 0，则 x + 3 = 0 或 x + 4 = 0。由此得到解 x = −3 和 x = −4。

When the coefficient of x² is not 1, the process requires more care. For example, 2x² + 5x + 2 = 0 can be factorised as (2x + 1)(x + 2) = 0, giving x = −½ and x = −2.

当 x² 的系数不为 1 时，过程需要更加细心。例如，2x² + 5x + 2 = 0 可分解为 (2x + 1)(x + 2) = 0，得到 x = −½ 和 x = −2。

Always check your factorisation by expanding the brackets. A quick expansion of (x + 3)(x + 4) confirms it returns x² + 7x + 12.

务必通过展开括号来检验因式分解是否正确。快速展开 (x + 3)(x + 4) 可以确认它还原为 x² + 7x + 12。

3. Solving by the Quadratic Formula | 求根公式法

Not all quadratic equations can be factorised easily. The quadratic formula works for every quadratic equation and is especially useful when roots are irrational or when you cannot spot the factors.

并非所有二次方程都能轻松因式分解。求根公式适用于每一个二次方程，在根为无理数或无法直接看出因式时尤其有效。

x = (−b ± √(b² − 4ac)) ÷ 2a

To use the formula, identify the values of a, b and c from the equation ax² + bx + c = 0, then substitute them carefully. Pay close attention to the signs of b and c.

使用公式时，先从方程 ax² + bx + c = 0 中识别 a、b、c 的值，然后仔细代入。要特别注意 b 和 c 的符号。

For example, solve 2x² + 3x − 5 = 0. Here a = 2, b = 3 and c = −5. Substituting into the formula gives x = (−3 ± √(9 + 40)) ÷ 4 = (−3 ± 7) ÷ 4.

例如，求解 2x² + 3x − 5 = 0。这里 a = 2，b = 3，c = −5。代入公式得 x = (−3 ± √(9 + 40)) ÷ 4 = (−3 ± 7) ÷ 4。

This yields two solutions: x = (−3 + 7) ÷ 4 = 1 and x = (−3 − 7) ÷ 4 = −2.5. Always simplify your final answers completely.

由此得到两个解：x = (−3 + 7) ÷ 4 = 1 和 x = (−3 − 7) ÷ 4 = −2.5。始终将最终答案化为最简形式。

Write the formula clearly in your exam paper. Examiners award method marks, so showing the substituted formula helps you earn credit even if your arithmetic goes wrong.

在试卷中清晰地写出公式。考官会给予方法分，因此即使计算有误，写出代入后的公式也能帮助你获得分数。

4. The Discriminant: b² − 4ac | 判别式：b² − 4ac

The discriminant, written as Δ = b² − 4ac, tells us how many real roots a quadratic equation has without solving it completely.

判别式写作 Δ = b² − 4ac，它能在不完全求解的情况下告诉我们二次方程有多少个实数根。

If Δ > 0, the equation has two distinct real roots. The graph crosses the x-axis at two separate points.

若 Δ > 0，方程有两个不同的实数根。图像在 x 轴上有两个不同的交点。

If Δ = 0, the equation has exactly one real root, often called a repeated or equal root. The graph just touches the x-axis at its vertex.

若 Δ = 0，方程只有一个实数根，通常称为重根或等根。图像在其顶点处恰好与 x 轴相切。

If Δ < 0, the equation has no real roots. The graph does not intersect the x-axis at all, and the solutions are complex numbers.

若 Δ < 0，方程没有实数根。图像完全不与 x 轴相交，解为复数。

For the equation x² − 6x + 9 = 0, the discriminant is 36 − 36 = 0, confirming one repeated root, x = 3. This is a classic exam question.

对于方程 x² − 6x + 9 = 0，判别式为 36 − 36 = 0，确认有一个重根 x = 3。这是一个经典的考试题目。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This form is extremely useful for finding the vertex of a parabola and solving equations directly.

配方法将二次式改写为 a(x + p)² + q 的形式。这种形式对求抛物线顶点和直接解方程极为有用。

Start with x² + 6x + 5 = 0. Take half of 6, which is 3, and write (x + 3)². Since (x + 3)² = x² + 6x + 9, we must subtract 9 to keep the expression equivalent.

从 x² + 6x + 5 = 0 开始。取 6 的一半，即 3，写成 (x + 3)²。因为 (x + 3)² = x² + 6x + 9，我们必须减去 9 以保持表达式等价。

Thus x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4. The equation becomes (x + 3)² − 4 = 0.

因此 x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。方程变为 (x + 3)² − 4 = 0。

Now solve: (x + 3)² = 4, so x + 3 = ±2. This gives x = −1 or x = −5. This method avoids factorisation entirely and works for all quadratics with real roots.

现在求解：(x + 3)² = 4，所以 x + 3 = ±2。由此得 x = −1 或 x = −5。这种方法完全绕开因式分解，适用于所有具有实数根的二次方程。

When the coefficient of x² is not 1, factor it out first. For 2x² − 8x + 3, write it as 2(x² − 4x) + 3, then complete the square inside the bracket.

当 x² 的系数不为 1 时，先将其提出。对于 2x² − 8x + 3，写成 2(x² − 4x) + 3，再在括号内配方。

6. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. The sign of a determines its orientation: if a > 0, the parabola opens upwards like a smile; if a < 0, it opens downwards like a frown.

y = ax² + bx + c 的图像是一条抛物线。a 的正负决定其开口方向：若 a > 0，抛物线开口向上，如同微笑；若 a < 0，开口向下，如同皱眉。

To sketch a quadratic graph, you need three key features: the y-intercept, the roots (if they exist), and the vertex. The y-intercept is simply c, the constant term.

要绘制二次函数草图，你需要三个关键特征：y 截距、根（如果存在）以及顶点。y 截距就是常数项 c。

For y = x² − 2x − 3, the y-intercept is −3. Factorising gives y = (x − 3)(x + 1), so the roots are x = 3 and x = −1.

对于 y = x² − 2x − 3，y 截距为 −3。因式分解得 y = (x − 3)(x + 1)，因此根为 x = 3 和 x = −1。

The vertex lies exactly halfway between the roots. Here the midpoint of 3 and −1 is 1, so the x-coordinate of the vertex is x = 1. Substitute back to find y = −4.

顶点恰好位于两个根的正中间。此处 3 和 −1 的中点为 1，因此顶点的 x 坐标为 x = 1。代回原式得 y = −4。

Mark the roots, the y-intercept and the vertex on a set of axes, then draw a smooth symmetric curve through them. Symmetry about the vertical line through the vertex is a key visual check.

在坐标轴上标出根、y 截距和顶点，然后绘制一条平滑的对称曲线穿过这些点。关于顶点所在竖直线的对称性是关键的视觉检验。

7. Roots, Vertex and Axis of Symmetry | 根、顶点与对称轴

For a quadratic function y = ax² + bx + c, the axis of symmetry is always the vertical line x = −b ÷ 2a. This line passes through the vertex and divides the parabola into two mirror images.

对于二次函数 y = ax² + bx + c，对称轴始终是竖直线 x = −b ÷ 2a。这条线穿过顶点，将抛物线分为左右对称的两部分。

The x-coordinate of the vertex is −b ÷ 2a. Substitute this value into the original equation to find the y-coordinate of the vertex.

顶点的 x 坐标为 −b ÷ 2a。将此值代入原方程即可求出顶点的 y 坐标。

In the previous example, y = x² − 2x − 3 has a = 1 and b = −2, so the axis of symmetry is x = 1. This matches our earlier midpoint calculation.

在前面的例子中，y = x² − 2x − 3 中 a = 1，b = −2，因此对称轴为 x = 1。这与我们之前的中点计算一致。

The vertex can also be read directly from the completed-square form. For y = (x + 3)² − 4, the vertex is (−3, −4), giving both coordinates instantly.

顶点还可以直接从配方法的形式中读出。对于 y = (x + 3)² − 4，顶点为 (−3, −4)，两个坐标一目了然。

Remember that the sum of the roots of ax² + bx + c = 0 is −b ÷ a and their product is c ÷ a. These relationships are handy shortcuts for checking your answers.

记住，方程 ax² + bx + c = 0 的两根之和为 −b ÷ a，两根之积为 c ÷ a。这些关系是检验答案的便捷捷径。

8. Quadratic Inequalities | 二次不等式

Quadratic inequalities extend the equation-solving skills to ranges of values. A typical question asks you to solve x² − 5x + 6 > 0 or x² − 5x + 6 ≤ 0.

二次不等式将方程求解的技能扩展到取值范围。典型问题要求解 x² − 5x + 6 > 0 或 x² − 5x + 6 ≤ 0。

First factorise: x² − 5x + 6 = (x − 2)(x − 3). The critical values, where the expression equals zero, are x = 2 and x = 3.

首先因式分解：x² − 5x + 6 = (x − 2)(x − 3)。表达式为零的关键值为 x = 2 和 x = 3。

Sketch the parabola mentally: it opens upwards, so the graph is above the x-axis for x < 2 and for x > 3. Thus x² − 5x + 6 > 0 means x < 2 or x > 3.

在脑海中画出抛物线草图：它开口向上，因此当 x < 2 和 x > 3 时图像在 x 轴上方。所以 x² − 5x + 6 > 0 的解为 x < 2 或 x > 3。

For x² − 5x + 6 ≤ 0, the graph lies on or below the x-axis between the roots, giving 2 ≤ x ≤ 3. Always check the inequality symbol carefully: strict symbols use open circles on a number line.

对于 x² − 5x + 6 ≤ 0，图像位于 x 轴下方或与 x 轴相交于两根之间，因此解为 2 ≤ x ≤ 3。务必仔细检查不等式符号：严格不等号在数轴上用空心圆圈表示。

Never multiply or divide an inequality by a negative number without reversing the inequality sign. This is a common source of lost marks.

切勿在不对换不等式方向的情况下将不等式乘以或除以负数。这是常见失分点。

9. Word Problems Leading to Quadratics | 二次方程应用题

Many exam questions require you to translate a real-world situation into a quadratic equation. Common contexts include area problems, projectile motion and number puzzles.

许多考试题目要求你将现实情境转化为二次方程。常见背景包括面积问题、抛体运动和数字谜题。

For example: a rectangle has length (x + 4) cm and width x cm, with area 60 cm². This gives x(x + 4) = 60, expanding to x² + 4x − 60 = 0.

例如：一个矩形长为 (x + 4) cm，宽为 x cm，面积为 60 cm²。这给出 x(x + 4) = 60，展开得 x² + 4x − 60 = 0。

Factorise into (x + 10)(x − 6) = 0, giving x = −10 or x = 6. Since a length cannot be negative, we reject x = −10 and take x = 6.

分解为 (x + 10)(x − 6) = 0，得 x = −10 或 x = 6。由于长度不能为负，我们舍去 x = −10，取 x = 6。

In the context of problem solving, always check whether both solutions are valid. Negative lengths, times or distances are meaningless in most real-world situations.

在应用题中，务必检验两个解是否都成立。在大多数现实情境中，负长度、负时间或负距离都是没有意义的。

Write your final answer as a full sentence that directly answers the question asked. This demonstrates clear communication and secures the final marks.

用完整的句子写出最终答案，直接回应问题所问。这展示了清晰的表达能力，并能确保拿到最后一部分分数。

10. Common Mistakes and How to Avoid Them | 常见错误与规避方法

One frequent error is forgetting to rearrange the equation into standard form before solving. Always write ax² + bx + c = 0 first.

一个常见错误是忘记在求解前将方程整理为标准形式。务必先写出 ax² + bx + c = 0。

Another mistake is misreading signs when using the quadratic formula. The term −b changes the sign of b, and the product 4ac must include the sign of c.

另一个错误是在使用求根公式时误读符号。−b 会改变 b 的符号，而 4ac 的乘积必须包含 c 的符号。

Students also forget that a quadratic equation must have two answers. If your solution gives only one value, check whether the discriminant is zero or whether you missed the ± symbol.

学生还会忘记二次方程必须有两个答案。如果你的解只有一个值，请检查判别式是否为零，或者你是否遗漏了 ± 符号。

When sketching graphs, always label the axes, plot the roots and vertex clearly, and make the parabola symmetric. Avoid drawing sharp corners at the vertex.

绘制图像时，务必标注坐标轴，清晰标出根和顶点，并确保抛物线对称。避免在顶点处画出尖角。

Finally, always check your solutions by substituting them back into the original equation. A quick check can catch arithmetic slips and save valuable marks.

最后，务必通过将解代回原方程来检验。快速检验可以捕捉计算失误并挽回应有的分数。

11. Practice Questions | 练习题

Question 1: Solve x² − 9x + 20 = 0 by factorisation. | 第 1 题：用因式分解法解 x² − 9x + 20 = 0。

Answer: (x − 4)(x − 5) = 0, so x = 4 or x = 5

Question 2: Use the quadratic formula to solve 3x² + 5x − 2 = 0. | 第 2 题：用求根公式解 3x² + 5x − 2 = 0。

Answer: x = (−5 ± √(25 + 24)) ÷ 6 = (−5 ± 7) ÷ 6, so x = ⅓ or x = −2

Question 3: Write y = x² − 4x + 7 in the form (x + p)² + q and state the vertex. | 第 3 题：将 y = x² − 4x + 7 写成 (x + p)² + q 的形式，并指出顶点。

Answer: y = (x − 2)² + 3, vertex at (2, 3)

Question 4: Determine the number of real roots of 2x² + 4x + 5 = 0 using the discriminant. | 第 4 题：用判别式判断 2x² + 4x + 5 = 0 的实数根个数。

Answer: Δ = 16 − 40 = −24 < 0, so no real roots

Try these questions on your own before reading the answers. Practising with pen and paper is the best way to build speed and accuracy.

先独立尝试这些题目，再看答案。用纸笔练习是提升速度与准确率的最佳方式。

12. Summary and Revision Tips | 总结与复习建议

Quadratic equations can be solved by factorisation, the quadratic formula or completing the square. Master all three methods and know when to use each one.

二次方程可以通过因式分解、求根公式或配方法求解。掌握这三种方法，并知道何时使用哪一种。

The discriminant b² − 4ac instantly reveals the number of real roots. Use it to check your answers and to guide your choice of method.

判别式 b² − 4ac 能立即揭示实数根的个数。用它来检验答案，并指导你选择解法。

For graphs, remember the axis of symmetry x = −b ÷ 2a and the vertex. Sketching a quick parabola helps you visualise inequalities and word problems.

对于图像，记住对称轴 x = −b ÷ 2a 和顶点。快速绘制抛物线草图有助于你直观理解不等式和应用题。

Practice past paper questions under timed conditions. Quadratic equations appear in Paper 2, Paper 4 and sometimes in Paper 6, so consistent practice is essential.

在限时条件下练习历年真题。二次方程出现在 Paper 2、Paper 4，有时也出现在 Paper 6，因此持续练习至关重要。

Keep a revision notebook of your mistakes and revisit it weekly. The student who learns from errors outpaces the student who simply repeats exercises.

准备一个复习笔记本记录你的错误，并每周回顾。从错误中学习的学生，会远超单纯机械刷题的学生。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mastering Quadratic Equations | 掌握二次方程