Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratic equations appear in nearly every IGCSE Mathematics paper. Mastering the techniques to solve them is essential for success in algebra, coordinate geometry, and even calculus at higher levels.

二次方程几乎出现在每一份 IGCSE 数学试卷中。掌握其求解方法，对于代数、坐标几何乃至更高阶的微积分学习都至关重要。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which is why it is called ‘quadratic’.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。变量的最高次数为 2，因此被称为”二次”。

For example, 2x² + 5x − 3 = 0 is a quadratic equation, while 3x + 7 = 0 is linear because its highest power is 1.

例如，2x² + 5x − 3 = 0 是二次方程，而 3x + 7 = 0 是一次（线性）方程，因为其最高次数为 1。

A quadratic equation may have two distinct real roots, one repeated real root, or no real roots, depending on the value of a key expression called the discriminant.

二次方程可能有两个不同的实数根、一个重根，或者没有实数根，这取决于一个关键表达式——判别式的值。

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2. Expanding and Factorising | 展开与因式分解

Before solving, you must be fluent in expanding brackets and factorising quadratics. To expand (x + p)(x + q), multiply each term in the first bracket by each term in the second: (x + p)(x + q) = x² + (p + q)x + pq.

在求解之前，你必须熟练展开括号和因式分解二次式。展开 (x + p)(x + q) 时，将第一个括号中的每一项与第二个括号中的每一项相乘：(x + p)(x + q) = x² + (p + q)x + pq。

For example, (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15.

例如，(x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15。

Factorising is the reverse process: given x² + bx + c, find two numbers whose sum is b and whose product is c.

因式分解是相反的过程：给定 x² + bx + c，找到两个数，使其和为 b、积为 c。

For instance, to factorise x² − 2x − 15, look for two numbers that multiply to −15 and add to −2. These are 3 and −5, so x² − 2x − 15 = (x + 3)(x − 5).

例如，要因式分解 x² − 2x − 15，需找到两个相乘得 −15、相加得 −2 的数。这两个数是 3 和 −5，因此 x² − 2x − 15 = (x + 3)(x − 5)。

When the coefficient of x² is not 1, such as 2x² + 5x − 3, use the method of splitting the middle term: multiply a and c (2 × −3 = −6), find two numbers whose product is −6 and sum is 5 (6 and −1), then rewrite and factorise by grouping.

当 x² 的系数不为 1 时，例如 2x² + 5x − 3，可使用拆中项法：将 a 与 c 相乘（2 × −3 = −6），找到乘积为 −6、和为 5 的两个数（6 和 −1），然后改写并分组因式分解。

2x² + 5x − 3 = 2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (x + 3)(2x − 1)

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3. Solving by Factorisation | 用因式分解求解

The principle behind solving by factorisation is the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

用因式分解求解的依据是零积性质：如果两个表达式的乘积为零，那么其中至少有一个必须为零。

If AB = 0, then A = 0 or B = 0. This allows us to split a quadratic into two simple linear equations.

若 AB = 0，则 A = 0 或 B = 0。这使我们能将一个二次方程拆分为两个简单的一次方程。

Solve x² − 2x − 15 = 0. First factorise: (x + 3)(x − 5) = 0. Then set each bracket to zero: x + 3 = 0 gives x = −3, and x − 5 = 0 gives x = 5.

求解 x² − 2x − 15 = 0。首先因式分解：(x + 3)(x − 5) = 0。然后令每个括号为零：x + 3 = 0 得 x = −3；x − 5 = 0 得 x = 5。

Always rearrange the equation into the form ax² + bx + c = 0 before factorising. For example, x² = 7x − 10 becomes x² − 7x + 10 = 0 = (x − 2)(x − 5), giving x = 2 or x = 5.

在因式分解之前，务必将方程整理为 ax² + bx + c = 0 的形式。例如，x² = 7x − 10 应转化为 x² − 7x + 10 = 0 = (x − 2)(x − 5)，解得 x = 2 或 x = 5。

If the quadratic cannot be factorised easily, or if you want a method that always works, use completing the square or the quadratic formula.

如果二次式不易因式分解，或者你希望使用一种通用方法，可以使用配方法或求根公式。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + h)² + k. This is especially useful for finding turning points of graphs and solving equations that do not factorise.

配方法将二次式改写为 a(x + h)² + k 的形式。这在求图像顶点和求解不能因式分解的方程时特别有用。

Start with x² + bx. Take half of b, square it, and add and subtract this value. For x² + 6x: half of 6 is 3, and 3² = 9, so x² + 6x = (x + 3)² − 9.

从 x² + bx 开始。取 b 的一半，将其平方，然后加上并减去该值。对于 x² + 6x：6 的一半是 3，3² = 9，因此 x² + 6x = (x + 3)² − 9。

To solve x² + 6x + 4 = 0, first complete the square: (x + 3)² − 9 + 4 = 0, so (x + 3)² − 5 = 0.

要求解 x² + 6x + 4 = 0，先配方：(x + 3)² − 9 + 4 = 0，因此 (x + 3)² − 5 = 0。

(x + 3)² = 5 → x + 3 = ±√5 → x = −3 ± √5

Hence the two solutions are x = −3 + √5 and x = −3 − √5. The ± symbol reminds us that both the positive and negative square roots must be considered.

因此两个解为 x = −3 + √5 和 x = −3 − √5。符号 ± 提醒我们必须同时考虑正负平方根。

For a general quadratic ax² + bx + c, the completed square form is a[(x + b/(2a))² − (b² − 4ac)/(4a²)]. This leads directly to the quadratic formula.

对于一般二次方程 ax² + bx + c，配方法的形式为 a[(x + b/(2a))² − (b² − 4ac)/(4a²)]。这直接推导出求根公式。

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5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0 in one step. It is derived from completing the square and is stated as follows.

求根公式可以一步求解任意二次方程 ax² + bx + c = 0。它由配方法推导而来，表达式如下。

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, identify a, b and c carefully, substitute them in, and simplify. Note that a is the coefficient of x², b is the coefficient of x, and c is the constant term.

使用公式时，仔细确定 a、b、c，代入并化简。注意 a 是 x² 的系数，b 是 x 的系数，c 是常数项。

Solve 2x² + 5x − 3 = 0 using the formula. Here a = 2, b = 5, c = −3.

用求根公式解 2x² + 5x − 3 = 0。这里 a = 2，b = 5，c = −3。

x = (−5 ± √(5² − 4 × 2 × (−3))) / (2 × 2) = (−5 ± √(25 + 24)) / 4 = (−5 ± √49) / 4 = (−5 ± 7) / 4

This gives x = (−5 + 7)/4 = 2/4 = 1/2, and x = (−5 − 7)/4 = −12/4 = −3. The solutions are x = 1/2 and x = −3, which match the factorisation result from earlier.

由此得 x = (−5 + 7)/4 = 2/4 = 1/2，以及 x = (−5 − 7)/4 = −12/4 = −3。解为 x = 1/2 和 x = −3，与之前因式分解的结果一致。

You must memorise the quadratic formula for the IGCSE examination. It is not provided on the formula sheet in all papers, so practise it until you can write it instantly.

你必须牢记求根公式以备 IGCSE 考试之用。并非所有试卷的公式表都会提供它，因此要反复练习，直到能立即写出。

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6. The Discriminant | 判别式

The expression inside the square root in the quadratic formula, b² − 4ac, is called the discriminant. Its value determines the nature of the roots.

求根公式中根号内的表达式 b² − 4ac 称为判别式。其值决定根的性质。

• If b² − 4ac > 0: two distinct real roots. The graph crosses the x-axis at two points.
• 如果 b² − 4ac > 0：两个不同的实数根。图像与 x 轴相交于两点。
• If b² − 4ac = 0: one repeated real root. The graph touches the x-axis at one point (the vertex lies on the axis).
• 如果 b² − 4ac = 0：一个重根。图像与 x 轴相切于一点（顶点在轴上）。
• If b² − 4ac < 0: no real roots. The graph does not meet the x-axis at all.
• 如果 b² − 4ac < 0：没有实数根。图像与 x 轴无交点。

For x² − 2x − 15 = 0, the discriminant is (−2)² − 4(1)(−15) = 4 + 60 = 64, which is positive and a perfect square. This confirms two rational roots, and the equation can be factorised.

对于 x² − 2x − 15 = 0，判别式为 (−2)² − 4(1)(−15) = 4 + 60 = 64，为正且为完全平方数。这确认了有两个有理根，且方程可以因式分解。

When the discriminant is a perfect square, factorisation is possible; when it is positive but not a perfect square, the roots are irrational and you should use the quadratic formula.

当判别式为完全平方数时，可以因式分解；当判别式为正但不是完全平方数时，根为无理数，应使用求根公式。

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7. 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, forming a U-shape; if a < 0, it opens downward, forming an inverted U-shape.

y = ax² + bx + c 的图像是抛物线。若 a > 0，抛物线开口向上，呈 U 形；若 a < 0，开口向下，呈倒 U 形。

The roots of the equation ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis. Solving the equation is equivalent to finding these intercepts.

方程 ax² + bx + c = 0 的根就是图像与 x 轴交点的横坐标。求解方程等价于找到这些截距。

The vertex (turning point) can be found using the completed square form. For y = a(x + h)² + k, the turning point is (−h, k).

顶点（转折点）可以通过配方法形式求得。对于 y = a(x + h)² + k，顶点为 (−h, k)。

For example, y = (x + 3)² − 5 has its minimum point at (−3, −5). The axis of symmetry is the vertical line x = −3, which passes through the vertex.

例如，y = (x + 3)² − 5 在 (−3, −5) 处取得最小值。对称轴是经过顶点的竖直线 x = −3。

Alternatively, the x-coordinate of the vertex is −b/(2a) for y = ax² + bx + c. Substitute this back into the equation to find the y-coordinate.

另一种方法：对于 y = ax² + bx + c，顶点的横坐标为 −b/(2a)。将其代回方程即可得到纵坐标。

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8. Word Problems and Applications | 应用题与实际应用

Quadratic equations frequently appear in geometry, number problems and projectile motion. The key is to translate the problem into algebra, then solve using the appropriate method.

二次方程经常出现在几何、数字问题以及抛体运动中。关键在于将问题转化为代数形式，然后用适当的方法求解。

Example: The length of a rectangle is 3 cm longer than its width, and its area is 40 cm². Find the width.

例题：一个长方形的长比宽长 3 cm，面积为 40 cm²。求宽。

Let the width be x. Then the length is x + 3, and the area is x(x + 3) = 40. Expanding gives x² + 3x = 40, so x² + 3x − 40 = 0.

设宽为 x，则长为 x + 3，面积为 x(x + 3) = 40。展开得 x² + 3x = 40，即 x² + 3x − 40 = 0。

Factorise: (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a width cannot be negative, the width is 5 cm.

因式分解：(x + 8)(x − 5) = 0，得 x = −8 或 x = 5。由于宽度不能为负，所以宽为 5 cm。

When solving word problems, always check whether both answers make sense in the context. Negative lengths, negative times and negative distances are usually rejected.

在解应用题时，务必检查两个答案在情境中是否合理。负数长度、负数时间和负数距离通常应舍去。

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9. Common Mistakes | 常见错误

Below are the most frequent errors students make with quadratics, along with advice on how to avoid them.

以下是学生在处理二次方程时最常见的错误，以及如何避免这些错误的建议。

• Forgetting to rearrange to ax² + bx + c = 0. Always bring all terms to one side before factorising or using the formula.
• 忘记整理成 ax² + bx + c = 0。在因式分解或使用公式之前，务必将所有项移到同一边。
• Dividing both sides by x when x is a possible root. This loses the solution x = 0. For example, x² = 3x, dividing by x gives x = 3 but misses x = 0.
• 在 x 可能是根时两边同时除以 x。这会丢失解 x = 0。例如，x² = 3x 除以 x 得 x = 3，却漏掉了 x = 0。
• Sign errors in the quadratic formula. Pay special attention to −b and to c when c is negative. Use brackets when substituting.
• 求根公式中的符号错误。特别注意 −b 以及 c 为负数时的情况。代入时使用括号。
• Confusing the discriminant with the whole formula. The discriminant is only b² − 4ac, not the full denominator.
• 将判别式与整个公式混淆。判别式仅为 b² − 4ac，不包含分母部分。
• Forgetting the ± sign. The quadratic always has two solutions unless the discriminant is exactly zero.
• 忘记 ± 符号。除非判别式恰好为零，二次方程总有两个解。

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10. Summary and Practice | 总结与练习

To solve a quadratic equation, first try factorisation. If that fails, use the quadratic formula. Completing the square is useful for graph work and when an exact surd form is required.

求解二次方程时，首先尝试因式分解。若失败，则使用求根公式。配方法在图像分析或需要精确根式形式时非常有用。

The discriminant b² − 4ac tells you how many real roots exist, while the sign of a tells you whether the parabola opens upward or downward.

判别式 b² − 4ac 告诉你存在多少个实数根，而 a 的符号决定抛物线开口向上还是向下。

Method | 方法	Best Use | 最佳用途
Factorisation | 因式分解	When the discriminant is a perfect square | 判别式为完全平方数时
Completing the square | 配方法	Finding the vertex or solving with surd answers | 求顶点或根式解
Quadratic formula | 求根公式	Always works; ideal for non-factorisable equations | 通用方法；适用于不可因式分解的方程

Practise each method on the same equation to build confidence. For example, solve x² + 4x − 5 = 0 by all three methods and confirm that x = 1 and x = −5 in every case.

用相同方程练习每种方法以增强信心。例如，分别用三种方法解 x² + 4x − 5 = 0，并确认每种方法都得到 x = 1 和 x = −5。

Remember: check your solutions by substituting them back into the original equation. A solution is only correct if it satisfies the original statement.

记住：将解代回原方程进行验证。只有当解满足原方程时才是正确的。

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