📚 Quadratic Equations and Graphs | 二次方程与图像
A quadratic equation is one of the most central topics in IGCSE Mathematics. It appears in algebra, coordinate geometry, and applied problem-solving, so you need to be confident both with algebraic methods and with the shape and features of quadratic graphs.
二次方程是 IGCSE 数学中最核心的主题之一。它出现在代数、坐标几何和应用题中，因此你既要熟练掌握代数方法，也要理解二次函数图像的形状和关键特征。
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’.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此被称为“二次”方程。
For example, x² – 4 = 0, 2x² + 3x – 5 = 0 and x² = 9 are all quadratic equations. The condition a ≠ 0 is essential because if a = 0 the equation becomes linear, not quadratic.
例如，x² – 4 = 0、2x² + 3x – 5 = 0 和 x² = 9 都是二次方程。条件 a ≠ 0 至关重要，因为如果 a = 0，方程就变成了一次方程，而不是二次方程。
2. Standard Form and Terminology | 标准形式与术语
The standard form of a quadratic equation is written with all terms on one side and zero on the other:
二次方程的标准形式是将所有项移到一边，另一边为零：
ax² + bx + c = 0
Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. Before solving a quadratic equation, you should always rearrange it into standard form first.
其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。在求解二次方程之前，你应始终先将其整理为标准形式。
For example, 3x + x² = 5 should be rearranged to x² + 3x – 5 = 0. This makes it much easier to identify a, b and c for factorisation or when using the quadratic formula.
例如，3x + x² = 5 应整理为 x² + 3x – 5 = 0。这样可以更容易地识别 a、b 和 c，便于因式分解或使用二次公式。
3. Solving by Factorisation | 因式分解法求解
Factorisation is often the quickest method when the quadratic expression can be written as a product of two linear factors. The key idea is the zero product property: if p × q = 0, then p = 0 or q = 0.
当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。关键思想是零乘积性质：如果 p × q = 0，则 p = 0 或 q = 0。
The steps are: write the equation in standard form, factorise the quadratic expression, set each factor equal to zero, and solve the resulting linear equations.
步骤如下：将方程写成标准形式；对二次式进行因式分解；令每个因式等于零；解所得的一次方程。
For example, to solve x² – 7x + 12 = 0, factorise to get (x – 3)(x – 4) = 0. Then x – 3 = 0 gives x = 3, and x – 4 = 0 gives x = 4. The solutions are x = 3 and x = 4.
例如，解 x² – 7x + 12 = 0，因式分解得到 (x – 3)(x – 4) = 0。然后 x – 3 = 0 得 x = 3，x – 4 = 0 得 x = 4。解为 x = 3 和 x = 4。
4. Solving by Completing the Square | 配方法求解
Completing the square is a useful method when the quadratic does not factorise easily. It rewrites x² + bx as a perfect square plus or minus a constant using the identity:
当二次式不易因式分解时，配方法是一种非常有用的方法。它利用恒等式将 x² + bx 改写为一个完全平方加上或减去一个常数：
x² + bx = (x + b/2)² – (b/2)²
To solve an equation, move the constant term to the other side, add (b/2)² to both sides, write the perfect square, then take square roots.
解方程时，先将常数项移到另一边，两边加上 (b/2)²，写出完全平方，然后取平方根。
For example, to solve x² + 6x – 1 = 0, first write x² + 6x = 1. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 10, so (x + 3)² = 10. Taking square roots gives x + 3 = ±√10, so x = -3 ± √10.
例如，解 x² + 6x – 1 = 0，先写成 x² + 6x = 1。两边加上 (6/2)² = 9：x² + 6x + 9 = 10，因此 (x + 3)² = 10。取平方根得 x + 3 = ±√10，所以 x = -3 ± √10。
5. The Quadratic Formula | 二次公式
The quadratic formula provides a direct way to solve any quadratic equation in standard form. It is derived from completing the square and is given by:
二次公式提供了求解标准形式下任何二次方程的直接方法。它由配方法推导而来，公式如下：
x = (-b ± √(b² – 4ac)) ÷ (2a)
To use the formula, identify the coefficients a, b and c from ax² + bx + c = 0, substitute them into the formula, and simplify carefully.
使用该公式时，从 ax² + bx + c = 0 中识别系数 a、b 和 c，将其代入公式，并仔细化简。
For example, to solve 2x² + 3x – 2 = 0, we have a = 2, b = 3 and c = -2. Substituting gives x = (-3 ± √(9 + 16)) ÷ 4 = (-3 ± √25) ÷ 4 = (-3 ± 5) ÷ 4. Therefore x = 1/2 or x = -2.
例如，解 2x² + 3x – 2 = 0，我们有 a = 2、b = 3、c = -2。代入得 x = (-3 ± √(9 + 16)) ÷ 4 = (-3 ± √25) ÷ 4 = (-3 ± 5) ÷ 4。因此 x = 1/2 或 x = -2。
6. Discriminant and Nature of Roots | 判别式与根的性质
The expression under the square root in the quadratic formula, b² – 4ac, is called the discriminant. It tells us how many real roots a quadratic equation has without solving it fully.
二次公式中平方根内的表达式 b² – 4ac 被称为判别式。它可以在不完整求解的情况下告诉我们二次方程有多少个实根。
Denoted by Δ, the discriminant gives three cases for the equation ax² + bx + c = 0:
判别式用 Δ 表示，对于方程 ax² + bx + c = 0，它给出三种情况：
| Discriminant Δ = b² – 4ac | Nature of roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
For example, x² – 6x + 9 = 0 has Δ = 36 – 36 = 0, so it has one repeated root. Indeed the equation is (x – 3)² = 0, giving x = 3 only.
例如，x² – 6x + 9 = 0 的判别式 Δ = 36 – 36 = 0，因此它有一个重根。实际上该方程为 (x – 3)² = 0，仅得 x = 3。
At IGCSE level, if Δ < 0, you should write ‘no real roots’ because complex roots are not part of the course.
在 IGCSE 阶段，如果 Δ < 0，你应写“无实根”，因为复数根不在课程范围内。
7. Quadratic Graphs: Parabolas | 二次函数图像：抛物线
The graph of y = ax² + bx + c is a curve called a parabola. Its shape depends on the sign of a: if a > 0 the parabola opens upwards, and if a < 0 it opens downwards.
y = ax² + bx + c 的图像是一条称为抛物线的曲线。它的形状取决于 a 的符号：如果 a > 0，抛物线开口向上；如果 a < 0，抛物线开口向下。
The parabola has a vertical line of symmetry and a turning point called the vertex. The vertex is the minimum point when the parabola opens upwards and the maximum point when it opens downwards.
抛物线有一条竖直对称轴和一个称为顶点的转折点。当抛物线开口向上时，顶点是最低点；当开口向下时，顶点是最高点。
The y-intercept of y = ax² + bx + c is always (0, c). The x-intercepts, where the curve meets the x-axis, are the real solutions of ax² + bx + c = 0.
y = ax² + bx + c 的 y 轴截距始终为 (0, c)。曲线与 x 轴的交点（x 轴截距）就是方程 ax² + bx + c = 0 的实数解。
8. Key Features of a Parabola | 抛物线的关键特征
To sketch a quadratic graph accurately, you should identify its main features. The x-coordinate of the vertex is given by x = -b ÷ (2a), and the axis of symmetry is the vertical line with this same equation.
要准确地画出二次函数图像的草图，你应确定其主要特征。顶点的 x 坐标由 x = -b ÷ (2a) 给出，对称轴就是具有相同方程的竖直线。
Once you have the vertex x-coordinate, substitute it into the equation to find the y-coordinate of the vertex. The y-intercept is simply c.
得到顶点的 x 坐标后，将其代入方程即可求出顶点的 y 坐标。y 轴截距就是 c。
For example, for y = x² – 4x + 3, the vertex x-coordinate is x = 4 ÷ 2 = 2. Substituting x = 2 gives y = 4 – 8 + 3 = -1, so the vertex is (2, -1). The y-intercept is 3. Factorising gives x-intercepts at x = 1 and x = 3.
例如，对于 y = x² – 4x + 3，顶点 x 坐标为 x = 4 ÷ 2 = 2。代入 x = 2 得 y = 4 – 8 + 3 = -1，因此顶点为 (2, -1)。y 轴截距为 3。因式分解得到 x 轴截距为 x = 1 和 x = 3。
9. Solving Graphically | 图像法求解
Quadratic equations can also be solved graphically. The real roots of ax² + bx + c = 0 are the x-coordinates of the points where the parabola y = ax² + bx + c crosses the x-axis.
二次方程也可以通过图像法求解。方程 ax² + bx + c = 0 的实根就是抛物线 y = ax² + bx + c 与 x 轴交点的 x 坐标。
For example, the graph of y = x² – 2x – 3 crosses the x-axis at x = -1 and x = 3, so these are the solutions of x² – 2x – 3 = 0.
例如，y = x² – 2x – 3 的图像在 x = -1 和 x = 3 处与 x 轴相交，因此它们就是 x² – 2x – 3 = 0 的解。
Sometimes you may need to solve an equation like x² – 2x – 3 = 2. This can be done by drawing the same parabola and the horizontal line y = 2, then reading the x-coordinates of their intersection points.
有时你可能需要解像 x² – 2x – 3 = 2 这样的方程。这可以通过画出同一条抛物线和水平线 y = 2，然后读出它们交点的 x 坐标来完成。
10. Applications and Exam Tips | 应用与考试技巧
Quadratic equations often appear in real-life contexts such as finding unknown side lengths, consecutive number problems, or the height of a projectile. Translate the problem into a quadratic equation, solve it, and check that the answer makes sense in context.
二次方程经常出现在实际情境中，例如求未知边长、连续整数问题或物体的抛射高度。将问题转化为二次方程，求解后检查答案是否符合题意。
In exams, always rearrange into standard form before solving. Be careful with negative signs when using the quadratic formula, and remember that a negative discriminant means no real roots.
在考试中，求解前务必整理为标准形式。使用二次公式时要小心负号，并记住负判别式意味着没有实根。
After finding your solutions, substitute them back into the original equation to check they work. This is especially important when you have factorised or used the formula under time pressure.
求出解后，将它们代回原方程验证是否正确。在时间紧张的情况下进行因式分解或使用公式时，这一点尤为重要。
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