📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程完全掌握
Quadratic equations are at the heart of the IGCSE Mathematics syllabus. They appear in algebra, coordinate geometry, problem solving and even in optimisation questions. A confident understanding of quadratic equations will help you tackle a wide range of exam questions, from simple factorising to interpreting parabolas.
二次方程是 IGCSE 数学课程的核心内容。它们出现在代数、坐标几何、问题解决甚至优化题中。扎实掌握二次方程能帮助你应对从简单因式分解到解释抛物线等各种考试题型。
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 unknown x is 2, and this is what makes the equation quadratic. If a = 0, the equation becomes linear because the x² term disappears.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数，且 a ≠ 0。未知数 x 的最高次数是 2，这正是方程被称为二次的原因。如果 a = 0，方程就变成了一次方程，因为 x² 项消失了。
For example, x² – 7x + 10 = 0 and 3x² = 5x – 2 are both quadratic equations. The second equation needs rearranging before it is in standard form.
例如，x² – 7x + 10 = 0 和 3x² = 5x – 2 都是二次方程。第二个方程需要先整理才能变成标准形式。
2. Standard Form and Coefficients | 标准形式与系数
The standard form of a quadratic equation is:
ax² + bx + c = 0, where a ≠ 0
二次方程的标准形式为：
ax² + bx + c = 0，其中 a ≠ 0
The coefficient a is called the quadratic coefficient, b is the linear coefficient, and c is the constant term. Before solving a quadratic equation, always rearrange it into standard form. This makes factorising, completing the square and using the formula much easier.
系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。在解二次方程之前，务必先将其整理成标准形式。这会让因式分解、配方法和使用求根公式变得更加简单。
For example, 2x² – 8 = 0 can be written as 2x² + 0x – 8 = 0, so a = 2, b = 0 and c = -8.
例如，2x² – 8 = 0 可以写成 2x² + 0x – 8 = 0，因此 a = 2，b = 0，c = -8。
3. Solving by Factorising | 因式分解法
When a quadratic expression can be factorised into two linear brackets, we can set each bracket equal to zero. For example, x² + 5x + 6 = 0 can be written as (x + 2)(x + 3) = 0. This gives x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3.
当一个二次表达式可以因式分解为两个一次因式时，我们可以令每个括号等于零。例如，x² + 5x + 6 = 0 可以写成 (x + 2)(x + 3) = 0。于是得到 x + 2 = 0 或 x + 3 = 0，所以 x = -2 或 x = -3。
Factorising is often the fastest method, but it only works neatly when the roots are rational numbers. Always check your factorisation by expanding the brackets back to the original expression.
因式分解通常是最快的方法，但它只适用于根为有理数的情况。务必通过把括号展开回原式来检查因式分解是否正确。
4. Solving by Completing the Square | 配方法
Completing the square transforms a quadratic into the form (x + p)² = q. Start with x² + bx + c = 0, move c to the other side, then add (b/2)² to both sides. For example, x² + 6x + 5 = 0 becomes x² + 6x = -5, then add (6/2)² = 9 to get:
(x + 3)² = 4
配方法将二次方程转化为 (x + p)² = q 的形式。从 x² + bx + c = 0 开始，把 c 移到等号另一边，然后两边加上 (b/2)²。例如，x² + 6x + 5 = 0 变为 x² + 6x = -5，再加上 (6/2)² = 9，得到：
(x + 3)² = 4
Taking square roots gives x + 3 = ±2, so x = -1 or x = -5. This method is especially important when a quadratic cannot be factorised easily, and it also helps find the vertex of a parabola.
开平方得 x + 3 = ±2，所以 x = -1 或 x = -5。这种方法在二次方程不易因式分解时尤为重要，同时也有助于求抛物线的顶点。
5. The Quadratic Formula | 求根公式
The quadratic formula works for every quadratic equation. If ax² + bx + c = 0, then:
x = (-b ± √(b² – 4ac)) / (2a)
求根公式适用于所有二次方程。如果 ax² + bx + c = 0，则：
x = (-b ± √(b² – 4ac)) / (2a)
You simply substitute the values of a, b and c from the standard form and simplify. Always check the sign of each coefficient carefully, especially when b is negative.
只需将标准形式中的 a、b 和 c 的值代入并化简即可。务必仔细检查每个系数的符号，尤其是当 b 为负数时。
6. The Discriminant | 判别式
The expression under the square root, b² – 4ac, is called the discriminant and is often denoted by Δ. It tells us how many real roots a quadratic equation has.
平方根下的表达式 b² – 4ac 称为判别式，通常记作 Δ。它告诉我们二次方程有多少个实数根。
If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots, because the square root of a negative number is not real.
如果 Δ > 0，有两个不同的实数根。如果 Δ = 0，有一个重根。如果 Δ < 0，没有实数根，因为负数的平方根不是实数。
7. Graphical Interpretation: Parabolas | 图像解释：抛物线
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，并有一个最低点。如果 a < 0，抛物线开口向下，并有一个最高点。
The y-intercept is the constant term c, because when x = 0, y = c. The shape and position of the parabola depend on all three coefficients a, b and c.
y 轴截距是常数项 c，因为当 x = 0 时，y = c。抛物线的形状和位置取决于 a、b 和 c 三个系数。
8. Roots and x-Intercepts | 根与 x 轴交点
The solutions of the equation ax² + bx + c = 0 are exactly the x-intercepts of the graph y = ax² + bx + c. If the discriminant is positive, the graph cuts the x-axis at two points. If it is zero, the graph touches the x-axis at one point. If it is negative, the graph does not meet the x-axis at all.
方程 ax² + bx + c = 0 的解恰好就是图像 y = ax² + bx + c 与 x 轴的交点。如果判别式为正，图像与 x 轴相交于两点。如果判别式为零，图像与 x 轴相切于一点。如果判别式为负，图像与 x 轴没有交点。
This connection between algebra and graphs is tested frequently in IGCSE, so be ready to move between the equation and its graph.
代数与图像之间的这种联系在 IGCSE 中经常考查，因此要能够在方程与其图像之间灵活转换。
9. Vertex and Axis of Symmetry | 顶点与对称轴
A parabola has a vertex and a vertical axis of symmetry. Completing the square helps us find them: if y = a(x – h)² + k, the vertex is (h, k) and the axis of symmetry is x = h.
抛物线有一个顶点和一条竖直对称轴。配方法可以帮助我们找到它们：如果 y = a(x – h)² + k，则顶点为 (h, k)，对称轴为 x = h。
y = a(x – h)² + k
The vertex represents the maximum or minimum value of the quadratic function. For a > 0, k is the minimum value; for a < 0, k is the maximum value.
顶点代表二次函数的最大值或最小值。当 a > 0 时，k 是最小值；当 a < 0 时，k 是最大值。
10. Applications and Word Problems | 应用题与文字题
IGCSE questions often hide quadratic equations in real-life contexts such as area, projectile motion, profit and geometry. For example, a rectangle has length x + 3 and width x – 2, with area 20. Form the equation (x + 3)(x – 2) = 20, expand and solve the resulting quadratic.
IGCSE 考题常常把二次方程隐藏在现实情境中，例如面积、抛体运动、利润和几何。例如，一个长方形的长为 x + 3，宽为 x – 2，面积为 20。建立方程 (x + 3)(x – 2) = 20，展开并解所得的二次方程。
Always interpret your answer in the context of the problem and discard invalid solutions, such as negative lengths or negative time.
务必结合题意解释答案，并舍去无效解，例如负的长度或负的时间。
11. Common Mistakes | 常见错误
Students often forget to rearrange the equation into standard form before applying a method. They may divide both sides by x and lose a root, or forget the ± sign when taking square roots.
学生常常忘记在应用方法前先把方程整理成标准形式。他们可能会两边同除以 x 而丢掉一个根，或者在开平方时忘记写 ± 号。
Another common error is misidentifying the sign of b in the quadratic formula, especially when b is negative. Always check solutions by substituting them back into the original equation.
另一个常见错误是在求根公式中误判
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