Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像

📚 Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像

A quadratic equation is one of the most important algebraic topics in IGCSE Mathematics. It appears in solving equations, drawing graphs, working with inequalities and modelling real-life problems. This article covers the standard form, factorising, the quadratic formula, completing the square, the discriminant, graphs and common exam mistakes.

二次方程是 IGCSE 数学中最重要的代数主题之一。它出现在解方程、画图像、处理不等式以及建立实际问题模型等题型中。本文涵盖标准形式、因式分解法、求根公式法、配方法、判别式、图像以及常见考试错误。

1. Recognising Quadratic Equations | 识别二次方程

A quadratic equation in one variable is an equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear rather than quadratic. The highest power of x is 2.

一元二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。若 a = 0，方程就变成了一次方程。未知数 x 的最高次数为 2。

In IGCSE questions, you may need to rearrange a quadratic into standard form before solving. For example, x² = 5x − 6 becomes x² − 5x + 6 = 0. Always collect all terms on one side and set the expression equal to zero.

在 IGCSE 题目中，你可能需要先把二次方程整理成标准形式再求解。例如 x² = 5x − 6 可化为 x² − 5x + 6 = 0。一定要把所有项移到一边，并让表达式等于 0。

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2. Coefficients and Terminology | 系数与术语

In ax² + bx + c = 0, a is the leading coefficient, b is the coefficient of x, and c is the constant term. The sign of each term is part of the coefficient.

在 ax² + bx + c = 0 中，a 是二次项系数，b 是一次项系数，c 是常数项。每一项的正负号属于系数的一部分。

For 2x² − 3x + 4 = 0, we have a = 2, b = −3 and c = 4. For −x² + 2x = 0, we have a = −1, b = 2 and c = 0. Misreading these signs is a common source of error in the quadratic formula.

对于 2x² − 3x + 4 = 0，a = 2，b = −3，c = 4。对于 −x² + 2x = 0，a = −1，b = 2，c = 0。在读系数时弄错正负号是使用求根公式时常见的错误来源。

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3. Solving by Factorising | 因式分解法

Factorising is often the quickest method when the quadratic has simple roots. Write the quadratic in the form (px + q)(rx + s) = 0, then set each bracket equal to zero.

当二次方程有较简单的根时，因式分解通常是最快的方法。把二次式写成 (px + q)(rx + s) = 0 的形式，然后令每个括号分别等于 0。

Example: Solve x² − 5x + 6 = 0. Factorise to (x − 2)(x − 3) = 0, so x = 2 or x = 3.

例如：解 x² − 5x + 6 = 0。因式分解为 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。

Another example: Solve 2x² + 5x − 3 = 0. Factorise to (2x − 1)(x + 3) = 0, giving x = 1/2 or x = −3. Always check by expanding the brackets to ensure the middle term and constant term are correct.

另一个例子：解 2x² + 5x − 3 = 0。因式分解为 (2x − 1)(x + 3) = 0，得到 x = 1/2 或 x = −3。一定要通过展开括号来检查中间项和常数项是否正确。

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4. Solving by the Quadratic Formula | 求根公式法

The quadratic formula works for every quadratic equation, including those that do not factorise neatly:

求根公式适用于所有二次方程，包括不能整齐因式分解的方程：

x = (−b ± √(b² − 4ac)) ÷ (2a)

In this formula, a, b and c come from the standard form ax² + bx + c = 0. Substitute carefully with signs, especially when b or c is negative.

公式中的 a、b、c 来自标准形式 ax² + bx + c = 0。代入时要特别注意正负号，尤其是当 b 或 c 为负数时。

Example: Solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Then x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4 = (2 ± √10) ÷ 2.

例如：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。则 x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4 = (2 ± √10) ÷ 2。

When the question asks for exact answers, leave them in surd form. If a decimal approximation is required, use your calculator only at the final step.

当题目要求精确答案时，答案应保留根号形式。如果需要近似小数，应在最后一步再使用计算器。

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5. Completing the Square | 配方法

Completing the square rewrites the quadratic in the form a(x + p)² + q. This form is useful for finding the turning point of a graph and for solving when factorising is difficult.

配方法把二次式写成 a(x + p)² + q 的形式。这种形式便于求图像顶点，也适用于难以因式分解的方程。

Example: x² + 6x + 5 = (x + 3)² − 4. To solve x² + 6x + 5 = 0, we get (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.

例如：x² + 6x + 5 = (x + 3)² − 4。解 x² + 6x + 5 = 0 时，得到 (x + 3)² = 4，于是 x + 3 = ±2，解得 x = −1 或 x = −5。

When a ≠ 1, factor out a from the first two terms before completing the square. For example, 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5.

当 a ≠ 1 时，要先对前两项提取 a，再完成配方。例如 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5。

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6. Discriminant and Nature of Roots | 判别式与根的性质

The discriminant is the expression inside the square root in the quadratic formula: Δ = b² − 4ac. It tells you how many real solutions the equation has.

判别式是求根公式中平方根内的表达式：Δ = b² − 4ac。它告诉我们方程有多少个实数解。

Discriminant | 判别式	Nature of roots | 根的性质
Δ > 0	Two distinct real roots | 两个不同实数根
Δ = 0	One repeated real root | 一个重根
Δ < 0	No real roots | 无实数根

Example: For x² − 4x + 4 = 0, Δ = (−4)² − 4(1)(4) = 0, so there is one repeated root x = 2.

例如：对于 x² − 4x + 4 = 0，Δ = (−4)² − 4(1)(4) = 0，因此有一个重根 x = 2。

Example: For x² + x + 1 = 0, Δ = 1² − 4(1)(1) = −3, which is less than zero, so the equation has no real roots. The graph of y = x² + x + 1 never crosses the x-axis.

例如：对于 x² + x + 1 = 0，Δ = 1² − 4(1)(1) = −3，小于 0，因此方程没有实数根。y = x² + x + 1 的图像不与 x 轴相交。

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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 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。若 a > 0，抛物线开口向上，有最低点；若 a < 0，开口向下，有最高点。

The y-intercept is c because when x = 0, y = c. The roots are the x-intercepts, where the graph cuts or touches the x-axis. A quadratic with no real roots will not cross the x-axis at all.

y 轴截距是 c，因为当 x = 0 时 y = c。方程的根就是 x 轴截距，即图像与 x 轴相交或相切的横坐标。没有实数根的二次函数图像不会与 x 轴相交。

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8. Turning Point and Axis of Symmetry | 顶点与对称轴

By completing the square, y = a(x + p)² + q gives the turning point at (−p, q). The axis of symmetry is the vertical line x = −p.

通过配方法，y = a(x + p)² + q 的顶点为 (−p, q)。对称轴是竖直直线 x = −p。

Example: y = (x − 3)² − 4 has a minimum at (3, −4) and axis of symmetry x = 3.

例如：y = (x − 3)² − 4 在 (3, −4) 处取得最小值，对称轴为 x = 3。

If the equation is in standard form, the x-coordinate of the turning point is x = −b ÷ (2a); substitute this into the function to find the y-coordinate. For y = x² − 6x + 5, the x-coordinate is 3, and the y-coordinate is (3)² − 6(3) + 5 = −4, so the turning point is (3, −4).

如果方程是标准形式，顶点的 x 坐标是 x = −b ÷ (2a)；将其代入函数即可求出 y 坐标。对于 y = x² − 6x + 5，x 坐标为 3，y 坐标为 (3)² − 6(3) + 5 = −4，因此顶点为 (3, −4)。

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9. Solving Quadratic Inequalities | 二次不等式

To solve ax² + bx + c > 0 or ax² + bx + c < 0, first find the roots, then sketch the parabola. The sign of the quadratic changes at the roots.

解 ax² + bx + c > 0 或 ax² + bx + c < 0 时，先求出根，再画抛物线草图。二次式的符号在根处发生改变。

Example: Solve x² − x − 6 < 0. Roots are x = −2 and x = 3. Since a > 0, the graph is below the x-axis between the roots, so −2 < x < 3.

例如：解 x² − x − 6 < 0。根为 x = −2 和 x = 3。由于 a > 0，图像在两根之间位于 x 轴下方，因此 −2 < x < 3。

If the inequality were x² − x − 6 > 0, the solution would be x < −2 or x > 3. Always pay attention to whether the inequality is strict or includes equality.

如果不等式是 x² − x − 6 > 0，则解为 x < −2 或 x > 3。要注意不等式是否包含等号。

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

Quadratic equations often arise from area, projectile motion, profit and product problems. Define the unknown quantity clearly and form an equation from the given conditions.

二次方程常出现在面积、抛体运动、利润和乘积问题中。要清楚地设未知量，并根据条件建立方程。

Example: The product of two consecutive integers is 42. Let the smaller integer be n, so n(n + 1) = 42, giving n² + n − 42 = 0. Factorising gives (n + 7)(n − 6) = 0, so n = 6 or n = −7.

例如：两个连续整数的积为 42。设较小的整数为 n，则 n(n + 1) = 42，得 n² + n − 42 = 0。因式分解为 (n + 7)(n − 6) = 0，所以 n = 6 或 n = −7。

In geometry, the area of a rectangle with length (x + 3) and width x might be given as 40, leading to x(x + 3) = 40. Solve this as x² + 3x − 40 = 0 and then reject any negative solution for a length.

在几何题中，长为 (x + 3)、宽为 x 的长方形面积可能为 40，因此列出 x(x +

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