Mastering Quadratic Equations | 掌握一元二次方程

📚 Mastering Quadratic Equations | 掌握一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in Paper 2 and Paper 4, in both non-calculator and calculator sections, and form the foundation for higher-level algebra.

一元二次方程是IGCSE数学中最重要的考点之一。它在Paper 2和Paper 4中都会出现，既可能出现在非计算器部分，也可能出现在计算器部分，并且是更高阶代数的基础。

1. What Is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation is a polynomial equation of degree 2. In one variable x, the general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

一元二次方程是最高次数为2的多项式方程。在单个变量x中，一般形式为 ax² + bx + c = 0，其中a、b、c为常数，且 a ≠ 0。

The highest power of x is 2, which is why the equation is called “quadratic”. Because the highest power is 2, a quadratic equation has at most two real solutions.

x的最高次数为2，因此称为“二次”方程。由于最高次数为2，一元二次方程至多有两个实数解。

2. The Standard Form | 标准形式

Before solving, always rearrange the equation into the standard form ax² + bx + c = 0, with all terms on one side. For example, 2x² = 5x + 3 must be rewritten as 2x² − 5x − 3 = 0.

在求解之前，应先将方程整理为标准形式 ax² + bx + c = 0，即把所有项移到等号一边。例如，2x² = 5x + 3 必须改写为 2x² − 5x − 3 = 0。

This step is essential because factorisation methods and the quadratic formula both assume the standard form. Always check the signs of b and c carefully.

这一步至关重要，因为因式分解法和求根公式都默认使用标准形式。务必仔细检查b和c的符号。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic factorises neatly. To solve x² − 5x + 6 = 0, find two numbers that multiply to +6 and add to −5: these are −2 and −3.

当二次式能整洁地分解时，因式分解是最快的方法。要解 x² − 5x + 6 = 0，找两个数相乘为+6、相加为−5：它们就是−2和−3。

Hence the equation becomes (x − 2)(x − 3) = 0. Using the zero product property, either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

因此方程变为 (x − 2)(x − 3) = 0。根据零乘积性质，x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

If ab = 0, then a = 0 or b = 0.

若 ab = 0，则 a = 0 或 b = 0。

When the coefficient of x² is not 1, use factor pairs carefully. For 2x² + 7x + 3 = 0, we write 2x² + 6x + x + 3 = 0, then factor by grouping: (2x + 1)(x + 3) = 0.

当x²的系数不为1时，要仔细使用因数对。对于 2x² + 7x + 3 = 0，可写为 2x² + 6x + x + 3 = 0，再分组分解得 (2x + 1)(x + 3) = 0。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. For x² + 6x + 2 = 0, take half of 6, which is 3, and write (x + 3)² − 9 + 2 = 0.

配方法将二次式改写为 a(x + p)² + q 的形式。对于 x² + 6x + 2 = 0，取6的一半为3，写成 (x + 3)² − 9 + 2 = 0。

This simplifies to (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, so x = −3 ± √7.

化简得 (x + 3)² = 7。两边开平方得 x + 3 = ±√7，所以 x = −3 ± √7。

Remember: when you take a square root, you must include both the positive and negative signs. The symbol ± represents both solutions.

记住：开平方时必须同时取正负两个符号。符号±表示两个解。

5. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation, including those that do not factorise. For ax² + bx + c = 0, the solutions are given by:

求根公式适用于所有一元二次方程，包括那些无法因式分解的方程。对于 ax² + bx + c = 0，解由下式给出：

x = (−b ± √(b² − 4ac)) / 2a

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substituting gives x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4.

例如，解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入得 x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4。

This simplifies to x = 1 ± √10/2. On a calculator paper, give your answer to 3 significant figures: x ≈ 2.58 or x ≈ −0.581.

化简得 x = 1 ± √10/2。在计算器试卷中，答案保留3位有效数字：x ≈ 2.58 或 x ≈ −0.581。

6. The Discriminant | 判别式

The discriminant is the expression Δ = b² − 4ac. It tells us the number and nature of the roots without solving the equation.

判别式是表达式 Δ = b² − 4ac。它无需解方程就能告诉我们根的数量和性质。

If Δ > 0, there are two distinct real roots.

若 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is exactly one repeated real root.

若 Δ = 0，方程有一个重根。
If Δ < 0, there are no real roots.

若 Δ < 0，方程没有实数根。

For example, for x² − 4x + 4 = 0, Δ = 16 − 16 = 0, so there is one repeated root x = 2.

例如，对于 x² − 4x + 4 = 0，Δ = 16 − 16 = 0，所以有一个重根 x = 2。

7. Solving Word Problems | 应用题

Quadratic equations often appear in geometry and problem-solving questions. A rectangle has area 30 cm² and its length is 4 cm longer than its width. Let the width be x, then x(x + 4) = 30.

一元二次方程常出现在几何和应用题中。一个矩形面积为30 cm²，长比宽长4 cm。设宽为x，则 x(x + 4) = 30。

Expanding gives x² + 4x − 30 = 0. Using the formula, x = (−4 ± √(16 + 120)) / 2 = (−4 ± √136) / 2 ≈ 3.83 or −7.83.

展开得 x² + 4x − 30 = 0。使用求根公式，x = (−4 ± √(16 + 120)) / 2 = (−4 ± √136) / 2 ≈ 3.83 或 −7.83。

Since a width cannot be negative, x ≈ 3.83 cm. Always reject negative or impossible solutions in context.

因为宽度不能为负，所以 x ≈ 3.83 cm。在实际问题中，一定要舍去负数或不可能的根。

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

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward.

y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，开口向下。

The solutions of ax² + bx + c = 0 are the x-intercepts of the parabola, where the curve crosses the x-axis. If Δ < 0, the parabola does not cross the x-axis at all.

ax² + bx + c = 0 的解就是抛物线与x轴的交点，即曲线穿过x轴的位置。若 Δ < 0，抛物线完全不与x轴相交。

The axis of symmetry is x = −b/2a, and the vertex lies on this line. Completing the square directly reveals the coordinates of the vertex as (−p, q).

对称轴为 x = −b/2a，顶点就在这条直线上。配方后可直接得到顶点坐标为 (−p, q)。

9. Common Mistakes | 常见错误

Many students forget to set the equation to zero before factorising. For example, solving x² = 5x by dividing both sides by x gives x = 5, losing the solution x = 0.

许多学生在因式分解前忘记把方程化为零的形式。例如，解 x² = 5x 时，如果两边同除以x，会得到 x = 5，从而丢失了 x = 0 这个解。

Another common error is missing the ± sign when taking square roots. Remember that x² = 9 has two solutions: x = 3 and x = −3.

另一个常见错误是开平方时漏掉±号。记住 x² = 9 有两个解：x = 3 和 x = −3。

Also, in the quadratic formula, be careful with negative values of b and c. Write out each substitution step to avoid sign errors.

此外，在求根公式中，要小心b和c为负值的情况。写出每一步代入过程，以避免符号错误。

10. Exam Tips | 考试技巧

In non-calculator papers, try factorisation first. If the quadratic does not factorise easily, use completing the square. In calculator papers, the quadratic formula is often the safest route.

在非计算器试卷中，先尝试因式分解。如果不容易分解，就用配方法。在计算器试卷中，求根公式往往最稳妥。

Always check your solutions by substituting them back into the original equation. Also, give answers to the degree of accuracy requested, usually 3 significant figures or 2 decimal places.

务必把解代回原方程检验。同时，答案要按题目要求的精度给出，通常是3位有效数字或2位小数。

Finally, when a question asks for exact answers, leave surds in the form a ± b√c. Do not round unless instructed.

最后，当题目要求精确答案时，应保留根式形式 a ± b√c。除非题目要求，否则不要四舍五入。

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Mastering Quadratic Equations | 掌握一元二次方程