Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, graphs, word problems and even coordinate geometry. This revision guide will help you master every examinable aspect of quadratics, from factorisation to the discriminant, with clear steps and worked examples.

二次方程是 IGCSE 数学考纲中最重要的考点之一。它出现在代数、函数图像、应用题乃至坐标几何中。本复习指南将帮助你全面掌握二次方程的所有考点，从因式分解到判别式，每一步都配有清晰的讲解和例题。

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

A quadratic equation is a polynomial equation of degree 2. Its general form is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which gives the equation its name ‘quadratic’.

二次方程是最高次数为 2 的多项式方程。它的一般形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数是 2，因此得名“二次”方程。

Examples of quadratic equations include x² − 5x + 6 = 0 and 2x² + 3x − 2 = 0. Note that b or c may be zero, but a can never be zero, because then the equation would become linear.

二次方程的例子包括 x² − 5x + 6 = 0 和 2x² + 3x − 2 = 0。注意 b 或 c 可以为零，但 a 绝不能为零，否则方程就变成一次方程了。

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

Before solving any quadratic equation, you must write it in standard form: ax² + bx + c = 0, with all terms on one side and zero on the other. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.

在解任何二次方程之前，必须先将其写成标准形式：ax² + bx + c = 0，即所有项移到一边，另一边为零。ax² 称为二次项，bx 称为一次项，c 称为常数项。

For example, the equation x² + 4x = 21 must be rearranged to x² + 4x − 21 = 0 before solving. This step is essential, because all solution methods below require the standard form.

例如，方程 x² + 4x = 21 必须先变形为 x² + 4x − 21 = 0 才能求解。这一步至关重要，因为下面介绍的所有解法都要求标准形式。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer factors. We look for two numbers whose product is ac and whose sum is b. For the simple case where a = 1, we find two numbers whose product is c and whose sum is b.

当二次方程具有简单的整数因子时，因式分解是最快的方法。我们需要找到两个数，其乘积为 ac，其和为 b。对于 a = 1 的简单情况，就是找两个数，乘积为 c，和为 b。

Solve x² − 5x + 6 = 0. We need two numbers whose product is 6 and sum is −5. Those numbers are −2 and −3. Hence (x − 2)(x − 3) = 0, so x = 2 or x = 3.

解 x² − 5x + 6 = 0。我们需要找两个数，乘积为 6，和为 −5。这两个数是 −2 和 −3。因此 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。

When a ≠ 1, factorisation by grouping is used. For example, solve 2x² + 7x + 3 = 0. Multiply a and c: 2 × 3 = 6. Find two numbers whose product is 6 and sum is 7: 1 and 6. Rewrite the middle term: 2x² + x + 6x + 3 = 0, then factorise in pairs: x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0. Therefore x = −½ or x = −3.

当 a ≠ 1 时，使用分组分解法。例如，解 2x² + 7x + 3 = 0。将 a 和 c 相乘：2 × 3 = 6。找到乘积为 6 且和为 7 的两个数：1 和 6。改写中间项：2x² + x + 6x + 3 = 0，然后分组因式分解：x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0。因此 x = −½ 或 x = −3。

If (px + q)(rx + s) = 0, then px + q = 0 or rx + s = 0.

若 (px + q)(rx + s) = 0，则 px + q = 0 或 rx + s = 0。

4. Solving by the Quadratic Formula | 公式法

The quadratic formula can solve any quadratic equation, even when factorisation is impossible. You should memorise it and know when to apply it. The formula is derived from completing the square, and it gives both roots directly.

公式法可以解任何二次方程，即使无法因式分解也能求解。你应该牢记公式并知道何时使用。该公式由配方法推导而来，可以直接给出两个根。

For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a)

对于 ax² + bx + c = 0，x = (−b ± √(b² − 4ac)) / (2a)

Solve 3x² − 5x − 2 = 0 using the formula. Here a = 3, b = −5, c = −2. Substitute into the formula: x = (5 ± √(25 − 4 × 3 × (−2))) / (2 × 3) = (5 ± √(25 + 24)) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6. Therefore x = (5 + 7)/6 = 2 or x = (5 − 7)/6 = −1/3.

用公式法解 3x² − 5x − 2 = 0。这里 a = 3，b = −5，c = −2。代入公式：x = (5 ± √(25 − 4 × 3 × (−2))) / (2 × 3) = (5 ± √(25 + 24)) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6。因此 x = (5 + 7)/6 = 2 或 x = (5 − 7)/6 = −1/3。

Always check that the value under the square root, b² − 4ac, is not negative before applying the formula. If it is negative, the equation has no real roots, as we will discuss in the section on the discriminant.

在使用公式前，务必检查根号下的值 b² − 4ac 是否为非负数。如果它是负数，则方程没有实数根，这将在判别式一节中讨论。

5. Solving by Completing the Square | 配方法

Completing the square transforms ax² + bx + c into the form a(x + h)² + k. This method is useful for finding turning points and for deriving the quadratic formula itself. For the case a = 1, we take half of b, square it, and adjust the constant.

配方法将 ax² + bx + c 变形为 a(x + h)² + k 的形式。这种方法在求顶点坐标和推导公式法时非常有用。对于 a = 1 的情况，我们取 b 的一半，平方后调整常数项。

Solve x² + 6x − 7 = 0 by completing the square. Half of 6 is 3, and 3² = 9. Rewrite the equation as (x + 3)² − 9 − 7 = 0, so (x + 3)² = 16. Taking the square root of both sides, x + 3 = ±4, hence x = 1 or x = −7.

用配方法解 x² + 6x − 7 = 0。6 的一半是 3，且 3² = 9。将方程改写为 (x + 3)² − 9 − 7 = 0，所以 (x + 3)² = 16。两边开平方，x + 3 = ±4，因此 x = 1 或 x = −7。

When a ≠ 1, first factor out a from the x² and x terms. For example, 2x² + 8x + 5 = 0 becomes 2(x² + 4x) + 5 = 0, then 2(x + 2)² − 8 + 5 = 0, giving 2(x + 2)² = 3, so (x + 2)² = 1.5, and x = −2 ± √1.5.

当 a ≠ 1 时，先将 a 从 x² 和 x 项中提出。例如，2x² + 8x + 5 = 0 变为 2(x² + 4x) + 5 = 0，然后 2(x + 2)² − 8 + 5 = 0，得到 2(x + 2)² = 3，所以 (x + 2)² = 1.5，因此 x = −2 ± √1.5。

6. The Discriminant | 判别式

The discriminant is the expression b² − 4ac found under the square root in the quadratic formula. Its sign tells us how many real roots a quadratic equation has, without actually solving it. This is a frequent exam question in IGCSE.

判别式是公式法中根号下的表达式 b² − 4ac。它的符号告诉我们二次方程有多少个实数根，而不需要实际求解。这是 IGCSE 考试中的常见题型。

If b² − 4ac > 0, the equation has two distinct real roots.

若 b² − 4ac > 0，方程有两个不相等的实数根。
If b² − 4ac = 0, the equation has exactly one real root (a repeated root).

若 b² − 4ac = 0，方程恰好有一个实数根（重根）。
If b² − 4ac < 0, the equation has no real roots.

若 b² − 4ac < 0，方程没有实数根。

For example, determine the nature of the roots of x² − 4x + 3 = 0. Here b² − 4ac = (−4)² − 4 × 1 × 3 = 16 − 12 = 4 > 0, so there are two distinct real roots. For x² − 4x + 4 = 0, the discriminant is (−4)² − 4 × 1 × 4 = 0, so there is one repeated root.

例如，判断 x² − 4x + 3 = 0 的根的情况。这里 b² − 4ac = (−4)² − 4 × 1 × 3 = 16 − 12 = 4 > 0，所以有两个不相等的实数根。对于 x² − 4x + 4 = 0，判别式为 (−4)² − 4 × 1 × 4 = 0，所以有一个重根。

7. Sum and Product of Roots | 根与系数的关系

For a quadratic equation ax² + bx + c = 0, the sum of the roots is −b/a and the product of the roots is c/a. These relationships allow you to form a quadratic equation from given roots, and they are often tested in extended IGCSE papers.

对于二次方程 ax² + bx + c = 0，两根之和为 −b/a，两根之积为 c/a。这些关系可以用来根据已知根构造二次方程，在 IGSCE 扩展卷中经常考查。

Sum of roots: α + β = −b/a, Product of roots: αβ = c/a

两根之和：α + β = −b/a，两根之积：αβ = c/a

If the roots of a quadratic equation are 3 and −2, then the sum is 1 and the product is −6. Hence the equation is x² − (sum)x + (product) = 0, i.e. x² − x − 6 = 0. Check by factorising: (x − 3)(x + 2) = 0 indeed gives roots 3 and −2.

如果某个二次方程的两个根是 3 和 −2，则两根之和为 1，两根之积为 −6。因此方程为 x² − (两根之和)x + (两根之积) = 0，即 x² − x − 6 = 0。验算：因式分解 (x − 3)(x + 2) = 0 确实得到根 3 和 −2。

8. Quadratic Graphs and the Vertex | 二次函数图像与顶点

The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upward and has a minimum point; when a < 0, it opens downward and has a maximum point. The turning point is called the vertex.

y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有最小值点；当 a < 0 时，抛物线开口向下，有最大值点。这个转折点称为顶点。

The x-coordinate of the vertex is x = −b/(2a). You can find the y-coordinate by substituting this x-value back into the equation. Alternatively, completing the square gives y = a(x − h)² + k, where the vertex is (h, k).

顶点的 x 坐标为 x = −b/(2a)。将该 x 值代回方程即可求出 y 坐标。另一种求法是通过配方得到 y = a(x − h)² + k，其中顶点为 (h, k)。

Axis of symmetry: x = −b/(2a)

对称轴：x = −b/(2a)

For the graph y = x² − 4x + 1, the vertex x-coordinate is x = 4/2 = 2, and the y-coordinate is 2² − 4(2) + 1 = −3. So the vertex is at (2, −3). The axis of symmetry is the vertical line x = 2.

对于抛物线 y = x² − 4x + 1，顶点的 x 坐标为 x = 4/2 = 2，y 坐标为 2² − 4(2) + 1 = −3。因此顶点为 (2, −3)。对称轴是垂直线 x = 2。

9. Roots, Intercepts and the Graph | 根、截距与图像

The roots of ax² + bx + c = 0 are the x-intercepts of the graph y = ax² + bx + c. This link between algebra and geometry means that solving a quadratic is equivalent to finding where the parabola crosses the x-axis.

ax² + bx + c = 0 的根就是抛物线 y = ax² + bx + c 与 x 轴的交点横坐标。这种代数与几何之间的联系意味着，解二次方程等价于求抛物线与 x 轴的交点。

The y-intercept of the graph is simply c, because when x = 0, y = c. To sketch the graph, you need the vertex, the y-intercept, and the x-intercepts if they exist. The discriminant tells you whether the parabola touches the x-axis or misses it entirely.

抛物线与 y 轴的交点就是 c，因为当 x = 0 时，y = c。要画抛物线草图，你需要顶点、y 轴截距以及 x 轴截距（如果存在）。判别式告诉你抛物线是与 x 轴相交、相切还是完全不相交。

If y = (x − 1)(x − 3), the roots are x = 1 and x = 3, the y-intercept is 3, and the vertex lies midway between the roots at x = 2, with y = (1)(−1) = −1. Sketch these key points and draw a smooth U-shaped curve.

如果 y = (x − 1)(x − 3)，则根为 x = 1 和 x = 3，y 轴截距为 3，顶点位于两根中点 x = 2 处，y = (1)(−1) = −1。标出这些关键点，画一条平滑的 U 形曲线即可。

10. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities look like ax² + bx + c > 0 or ax² + bx + c < 0. To solve them, first factorise the quadratic, then sketch the parabola or use a number line to decide which regions satisfy the inequality.

二次不等式形如 ax² + bx + c > 0 或 ax² + bx + c < 0。解这类不等式时，先因式分解二次式，然后画出抛物线草图或使用数轴来确定哪些区域满足不等式。

Solve x² − x − 6 > 0. Factorise to get (x − 3)(x + 2) > 0. The critical values are x = −2 and x = 3. Test intervals: for x < −2, both factors are negative, product positive, so inequality holds. For −2 < x < 3, one factor is negative, product negative, so inequality fails. For x > 3, both positive, product positive, so inequality holds.

解 x² − x − 6 > 0。因式分解得 (x − 3)(x + 2) > 0。临界值为 x = −2 和 x = 3。测试区间：当 x < −2 时，两个因子都为负，乘积为正，不等式成立。当 −2 < x < 3 时，一个因子为负，乘积为负，不等式不成立。当 x > 3 时，两个因子都为正，乘积为正，不等式成立。

x² − x − 6 > 0 ⇒ x < −2 or x > 3

x² − x − 6 > 0 ⇒ x < −2 或 x > 3

For the inequality x² − x − 6 < 0, the solution is the middle interval only: −2 < x < 3. Remember that the inequality symbol flips if you multiply or divide by a negative number, though this is rarely needed when solving by factorisation.

对于不等式 x² − x − 6 < 0，解只取中间区间：−2 < x < 3。请记住，如果乘以或除以负数，不等式符号要反转，不过在因式分解法中很少需要这样做。

11. Common Exam Mistakes | 常见考试错误

Many students lose marks in quadratic questions due to avoidable errors. The most common mistake is forgetting to rearrange the equation into standard form before factorising or applying the formula. Another is incorrectly expanding (x + 3)² as x² + 9 instead of x² + 6x + 9.

很多学生在二次方程题目中因为可以避免的错误而失分。最常见的错误是忘记先将方程整理成标准形式再进行因式分解或套用公式。另一个常见错误是将 (x + 3)² 错误展开为 x² + 9，而不是 x² + 6x + 9。

Always write the equation as ax² + bx + c = 0 first.

务必先将方程写成 ax² + bx + c = 0 的形式。
Do not cancel a variable from both sides if it may equal zero.

如果变量可能等于零，不要从两边约去该变量。
Check your answers by substituting back into the original equation.

通过代回原方程来检查答案。
Read the question carefully: are they asking for roots, the vertex, or the discriminant?

仔细审题：题目要求的是根、顶点还是判别式？
When using the quadratic formula, write down a, b and c explicitly before substituting.

使用公式法时，先写出 a、b、c 再代入。

Another frequent error in graph questions is confusing the direction of the parabola. Check the sign of a: a > 0 means a ‘smile’ shape, a < 0 means a 'frown' shape.

在图像题中另一个常见错误是混淆抛物线的开口方向。检查 a 的符号：a > 0 表示“微笑”型，a < 0 表示“皱眉”型。

12. Practice Questions | 练习题

Medicine is best learned by practice, and mathematics is no exception. Try these questions under exam conditions, then check your answers. If you make a mistake, review the relevant section of this guide before moving on.

医学靠实践，数学也不例外。请在考试条件下尝试这些题目，然后核对答案。如果做错了，先回顾本指南的相关章节再继续。

Question 1 | 练习 1: Solve x² − 7x + 10 = 0 by factorisation.

Question 1 | 练习 1: 用因式分解法解 x² − 7x + 10 = 0。

Question 2 | 练习 2: Solve 2x² + 5x − 3 = 0 using the quadratic formula.

Question 2 | 练习 2: 用公式法解 2x² + 5x − 3 = 0。

Question 3 | 练习 3: Find the vertex of y = x² − 6x + 5 and state the axis of symmetry.

Question 3 | 练习 3: 求 y = x² − 6x + 5 的顶点坐标，并写出对称轴方程。

Question 4 | 练习 4: Determine the number of real roots of x² + 4x + 5 = 0.

Question 4 | 练习 4: 判断 x² + 4x + 5 = 0 的实数根个数。

Answers: 1) x = 2 or x = 5. 2) x = ½ or x = −3. 3) Vertex (3, −4), axis x = 3. 4) Discriminant = 16 − 20 = −4 < 0, so no real roots.

答案：1) x = 2 或 x = 5。2) x = ½ 或 x = −3。3) 顶点 (3, −4)，对称轴 x = 3。4) 判别式 = 16 − 20 = −4 < 0，所以没有实数根。

Understanding quadratic equations opens the door to many other topics in IGCSE mathematics, including functions, inequalities and optimisation. Master the techniques above, practise regularly, and you will answer quadratic questions with confidence in your exam.

理解二次方程为你打开了 IGCSE 数学中许多其他话题的大门，包括函数、不等式和最值问题。掌握以上技巧，定期练习，你就能在考试中自信地解答二次方程相关题目。

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Mastering Quadratic Equations | 掌握二次方程