Solving Quadratic Equations: Methods and Graphs | 解二次方程：方法与图像

📚 Solving Quadratic Equations: Methods and Graphs | 解二次方程：方法与图像

A quadratic equation is a polynomial equation of degree 2. In its standard form it is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The equation is called “quadratic” because the highest power of the variable x is 2. In the IGCSE syllabus, solving quadratic equations is a fundamental skill that supports curve sketching, inequalities and many real-world problems.

二次方程是次数为2的多项式方程。其标准形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。之所以称为”二次”，是因为变量 x 的最高次数为2。在 IGCSE 考纲中，解二次方程是绘制曲线、解不等式和解决许多实际问题的基本技能。

At IGCSE level, you are expected to solve quadratic equations by three main methods: factorisation, the quadratic formula and completing the square. You must also be able to interpret the graph of a quadratic function and use its features to solve problems. This article introduces each method step by step, paired with worked examples and examination tips.

在 IGCSE 阶段，你须掌握三种解二次方程的主要方法：因式分解法、求根公式法和配方法。你还必须能够理解二次函数的图像，并利用其特征解决问题。本文将逐步介绍每种方法，并配以例题和考试技巧。

1. Standard Form and Key Terms | 标准形式与关键术语

Every quadratic equation should be rearranged into the standard form ax² + bx + c = 0 before solving. This form makes the coefficients easy to identify and is required for applying the quadratic formula. The table below summarises the key components of a quadratic expression.

每个二次方程在求解前都应整理成标准形式 ax² + bx + c = 0。这种形式便于识别系数，也是使用求根公式的前提。下表总结了二次表达式的关键组成部分。

Term \| 项	Meaning \| 含义
ax²	Quadratic term / 二次项
bx	Linear term / 一次项
c	Constant term / 常数项
Root / 根	Value of x that makes the equation equal to 0 / 使方程等于0的 x 值

A quadratic equation has at most two real roots. This is because a parabola intersects the x-axis at no, one or two points. Understanding this geometric idea will help you check whether your algebraic answers make sense.

二次方程至多有两个实数根。这是因为抛物线与 x 轴相交于0个、1个或2个点。理解这一几何意义有助于检验你的代数答案是否合理。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic expression can be written as a product of two linear factors. The key idea is the zero product property: if A × B = 0, then A = 0 or B = 0.

因式分解法是当二次表达式能写成两个一次因式的乘积时最快的方法。其核心思想是零乘积性质：若 A × B = 0，则 A = 0 或 B = 0。

Worked example: solve x² − 5x + 6 = 0.

例题：解方程 x² − 5x + 6 = 0。

We look for two numbers that multiply to give 6 and add to give −5. The numbers are −2 and −3, so the factorisation is (x − 2)(x − 3) = 0.

寻找两个数，它们的乘积为6，和为−5。这两个数是−2和−3，因此因式分解为 (x − 2)(x − 3) = 0。

(x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3

Always expand your factors to check the expansion matches the original expression. If the coefficient of x² is not 1, you may need to use a multiplication grid or trial and error, but the product-sum idea still applies.

展开因式时要检查结果是否与原表达式一致。若 x² 的系数不为1，可使用乘法网格或试错法，但积与和的思想仍然适用。

3. The Quadratic Formula | 求根公式

When factorisation is not obvious, the quadratic formula works for every quadratic equation. For the general equation ax² + bx + c = 0, the solutions are given by the formula below.

当因式分解不明显时，求根公式适用于所有二次方程。对于一般方程 ax² + bx + c = 0，解由以下公式给出。

x = (−b ± √(b² − 4ac)) / 2a

This formula is derived by completing the square on the general equation, and it always gives the exact roots when they exist. You must memorise it and use it carefully with the correct signs.

该公式通过对一般方程配方法推导而来，只要根存在，它总能给出精确解。你必须牢记该公式并注意符号的正确使用。

Worked example: solve 2x² − 3x − 5 = 0. Here a = 2, b = −3, c = −5.

例题：解方程 2x² − 3x − 5 = 0。这里 a = 2，b = −3，c = −5。

x = (3 ± √(9 + 40)) / 4 = (3 ± 7) / 4

This gives x = (3 + 7)/4 = 2.5, or x = (3 − 7)/4 = −1. Both values satisfy the original equation, so the solution set is {2.5, −1}.

由此得到 x = (3 + 7)/4 = 2.5，或 x = (3 − 7)/4 = −1。两个值均满足原方程，因此解集为 {2.5, −1}。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form (x + p)² + q. This method is particularly useful for finding turning points and solving equations without factorising. For a simple quadratic x² + bx + c, we use the following identity.

配方法将二次表达式改写成 (x + p)² + q 的形式。这种方法在求顶点坐标和解方程时特别有用，无需因式分解。对于简单的二次式 x² + bx + c，我们使用以下恒等式。

x² + bx + c = (x + b/2)² − (b/2)² + c

Worked example: solve x² + 6x + 2 = 0. Since half of 6 is 3 and 3² = 9, we rewrite the equation as (x + 3)² − 9 + 2 = 0, so (x + 3)² = 7.

例题：解方程 x² + 6x + 2 = 0。因为6的一半是3，且 3² = 9，所以方程改写为 (x + 3)² − 9 + 2 = 0，即 (x + 3)² = 7。

x + 3 = ±√7 ⇒ x = −3 ± √7

When a ≠ 1, first divide the entire equation by a. Then apply the same procedure to the resulting expression. Keeping the square isolated on one side makes the algebra tidy.

当 a ≠ 1 时，先将整个方程除以 a，再对所得表达式应用同样的步骤。将完全平方单独放在等式一侧，可以使代数运算更加整洁。

5. The Discriminant | 判别式

The discriminant is the expression Δ = b² − 4ac inside the quadratic formula. It tells us the nature of the roots without fully solving the equation.

判别式是求根公式中的表达式 Δ = b² − 4ac。它可以在不完整求解的情况下告诉我们根的性质。

Δ = b² − 4ac

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

若 Δ > 0，方程有两个不相等的实数根，图像与 x 轴相交于两个不同的点。
If Δ = 0, the equation has one repeated real root. The graph touches the x-axis at the vertex.

若 Δ = 0，方程有一个重根，图像与 x 轴相切于顶点。
If Δ < 0, the equation has no real roots. The graph never touches the x-axis.

若 Δ < 0，方程没有实数根，图像与 x 轴没有交点。

Check the discriminant first in an exam. If Δ is a perfect square, the roots are rational and factorisation is worth attempting; otherwise use the formula or complete the square.

考试中先判断判别式。若 Δ 是完全平方数，根为有理数，可优先尝试因式分解；否则使用求根公式或配方法。

6. Solving Quadratic Inequalities | 二次不等式

On the extended IGCSE paper, you may be asked to solve quadratic inequalities such as x² − 5x + 6 > 0. The method is to factorise first and sketch the parabola mentally.

在 IGCSE 扩展卷中，你可能需要解二次不等式，例如 x² − 5x + 6 > 0。方法是先因式分解，并在脑中画出抛物线草图。

Factorise to get (x − 2)(x − 3) > 0. The critical values are x = 2 and x = 3. Test the three intervals: x < 2, 2 < x < 3 and x > 3.

因式分解得 (x − 2)(x − 3) > 0。临界值为 x = 2 和 x = 3。测试三个区间：x < 2、2 < x < 3 和 x > 3。

x < 2 or x > 3

For inequalities with ≤ or ≥, include the critical values in the solution set. Remember that “less than 0” corresponds to the region between the roots, while “greater than 0” corresponds to the two outer regions.

对于 ≤ 或 ≥ 的不等式，解集中要包含临界值。注意”小于0″对应两根之间的区域，而”大于0″对应两个外侧区域。

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

The graph of y = ax² + bx + c is a parabola. Its shape depends only on the sign of a: if a > 0, the parabola opens upward like a U-shape and has a minimum point; if a < 0, it opens downward like an n-shape and has a maximum point.

函数 y = ax² + bx + c 的图像是抛物线。其开口方向仅取决于 a 的符号：若 a > 0，抛物线开口向上，呈 U 形，有最小值点；若 a < 0，抛物线开口向下，呈 n 形，有最大值点。

Key features of the graph include the y-intercept, always equal to c, and the x-intercepts, which are the roots of the equation. If Δ < 0, there are no x-intercepts; if Δ = 0, there is exactly one x-intercept.

图像的关键特征包括：y 轴截距始终等于 c；x 轴截距即方程的根。若 Δ < 0，则没有 x 轴截距；若 Δ = 0，则恰有一个 x 轴截距。

When sketching a quadratic, always label the axis of symmetry, the vertex and all intercepts. These labels are required for full marks on graph questions.

绘制二次函数图像时，一定要标注对称轴、顶点以及所有截距。在图像题中，这些标注是得满分所必需的。

8. Vertex and Axis of Symmetry | 顶点与对称轴

The axis of symmetry is a vertical line that divides the parabola into two mirror images. Its equation is given by the formula below.

对称轴是垂直于 x 轴的直线，它将抛物线分成左右对称的两部分。其方程由以下公式给出。

x = −b / (2a)

The vertex lies on this axis. To find the y-coordinate of the vertex, substitute x = −b/(2a) back into the original equation. Alternatively, if the quadratic is written in completed square form y = a(x − h)² + k, the vertex is simply (h, k).

顶点位于对称轴上。要求顶点的 y 坐标，只需将 x = −b/(2a) 代回原方程。另外，若二次式写成配方法形式 y = a(x − h)² + k，顶点坐标直接为 (h, k)。

This vertex represents the maximum or minimum value of the function. In optimisation problems, such as finding the greatest area or the highest point of a projectile, completing the square is usually the quickest approach.

顶点代表函数的最大值或最小值。在最优化问题中，例如求最大面积或抛射体的最高点，配方法通常是最快捷的方法。

9. Applications of Quadratics | 二次方程的应用

Quadratic equations appear frequently in IGCSE problem-solving questions. A classic example is a rectangle problem: the perimeter of a rectangle is 100 m and its area is 600 m². Let the length be x metres; then the width is 50 − x metres.

二次方程在 IGCSE 应用题中经常出现。一个经典例子是矩形问题：一个矩形的周长为100米，面积为600平方米。设长为 x 米，则宽为 50 − x 米。

x(50 − x) = 600 ⇒ x² − 50x + 600 = 0

Factorising gives (x − 20)(x − 30) = 0, so the sides are 20 m and 30 m. Many area, profit and motion problems reduce to a quadratic equation in exactly this way.

因式分解得 (x − 20)(x − 30) = 0，因此两边分别为20米和30米。许多面积、利润和运动问题都以这种方式归结为二次方程。

Another common application is projectile motion. If an object is thrown upward with height h = −5t² + 20t, the maximum height can be found by completing the square: h = −5(t − 2)² + 20, so the maximum height is 20 units at t = 2 seconds.

另一个常见应用是抛体运动。若物体上抛的高度为 h = −5t² + 20t，可通过配方法求最大高度：h = −5(t − 2)² + 20，因此在 t = 2 秒时达到最大高度20单位。

10. Working with Exact Surd Answers | 处理精确根式答案

When the discriminant is positive but not a perfect square, the roots involve square roots. In IGCSE, you should usually give your answer in exact surd form rather than a rounded decimal unless the question asks for a decimal.

当判别式为正数但不是完全平方数时，根会包含根号。在 IGCSE 考试中，除非题目特别要求小数，否则通常应给出精确的根式形式解答。

For example, solving x² + 4x + 1 = 0 using the formula gives x = −2 ± √3. Leave √3 as it is; converting to a decimal too early can introduce rounding errors in later parts of a question.

例如，用求根公式解 x² + 4x + 1 = 0 得到 x = −2 ± √3。保留√3不动；过早转换为小数可能会在后续问题中引入舍入误差。

Remember the rules of surds: √a × √b = √(ab), and √(a/b) = √a / √b. Simplifying the discriminant fully, for example √12 = 2√3, often makes the final answer cleaner.

记住根式运算法则：√a × √b = √(ab)，且 √(a/b) = √a / √b。将判别式中的根号完全化简，例如 √12 = 2√3，往往能让最终答案更简洁。

11. Common Exam Pitfalls | 常见考试陷阱

Forgetting the condition a ≠ 0. If a = 0, the equation is linear, not quadratic, and the formula does not apply.

忘记 a ≠ 0 的条件。若 a = 0，方程是一次方程而非二次方程，求根公式不适用。
Making sign errors when substituting negative values of b into the quadratic formula. Always bracket negative numbers.

将 b 的负值代入求根公式时出现符号错误。代入负数时始终加上括号。
Factorising without first rearranging to the form ax² + bx + c = 0. The zero product property only works when one side is exactly 0.

未先将方程整理为 ax² + bx + c = 0 就开始因式分解。零乘积性质只有在等式一侧恰为0时才成立。
Losing one root by dividing both sides by x. Division by x is only allowed if x ≠ 0, and you may remove a valid root.

两边同除以 x 导致丢根。除以 x 只有在 x ≠ 0 时才被允许，否则可能删掉一个有效根。
Mixing up the direction of an inequality sign when multiplying by a negative or when reading the solution from a graph.

乘以负数或从图像读取解集时弄错不等式方向。

Checking each answer by substitution into the original equation is the simplest way to catch these errors before you finish the paper.

将每个答案代回原方程进行检验，是完成试卷前发现这些错误的最简单方法。

12. Exam Strategies and Quick Checks | 应试策略与快速检验

Start by rearranging the equation into standard form. Then look at the discriminant: if it is a perfect square, try factorisation first; if not, use the quadratic formula or completing the square.

先将方程整理成标准形式，然后观察判别式：若判别式为完全平方数，优先尝试因式分解；若不是，则使用求根公式或配方法。

For graph questions, follow a fixed routine: find the axis of symmetry, then the vertex, then intercepts. This order is logical and prevents you from missing features. Always draw a quick sketch, even for algebraic questions, to visualise the situation.

做图像题时遵循固定步骤：先求对称轴，再求顶点，然后求截距。这个顺序逻辑清晰，可以避免遗漏特征。即使是代数题也建议画一个简图，以便直观理解问题。

Finally, read the question carefully to see whether you are asked for exact values or decimal approximations, and whether the domain of x is restricted. A correct method with the wrong precision can still lose marks.

最后，仔细阅读题目，确认要求的是精确值还是小数近似值，以及 x 的定义域是否有限制。方法正确但精度不符合要求同样可能丢分。

Mastering quadratics opens the door to many other IGCSE topics, including functions, sequences, calculus and coordinate geometry. Practise all three solving methods until they feel automatic, and your exam confidence will grow quickly.

掌握二次方程为学习 IGCSE 的许多其他主题打下基础，包括函数、数列、微积分和坐标几何。反复练习三种解法，直到运用自如，你的考试信心也会迅速提升。

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Solving Quadratic Equations: Methods and Graphs | 解二次方程：方法与图像