Quadratic Equations and Graphs | 二次方程与图像

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

A quadratic equation is one of the most fundamental topics in IGCSE Mathematics. It appears in algebra, graphing, and word problems, and it builds a bridge toward advanced functions. Understanding its forms, solution methods, and graph features will help you solve many exam-style questions with confidence.

二次方程是 IGCSE 数学中最基础的主题之一。它出现在代数、图像和文字题中，也为更高级的函数学习搭建桥梁。理解它的形式、解法和图像特征，将帮助你自信地解决许多考试题型。

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

A quadratic equation is a polynomial equation of degree two. It can always be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a must not be zero. If a were zero, the equation would become linear rather than quadratic.

二次方程是一个二次多项式方程。它总能写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不能为零。如果 a 为零，方程就会变成一次方程而不是二次方程。

ax² + bx + c = 0, a ≠ 0

For example, 2x² − 5x + 3 = 0 is quadratic because the highest power of x is 2. Here the coefficients are a = 2, b = −5 and c = 3.

例如，2x² − 5x + 3 = 0 是二次方程，因为 x 的最高次数是 2。这里系数为 a = 2，b = −5，c = 3。

2. Recognising Quadratic Graphs | 识别二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point. The curve is always symmetrical about a vertical line through its turning point.

y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上且有最小值点；如果 a < 0，则开口向下且有最大值点。曲线始终关于经过其顶点的竖直线对称。

The y-intercept is always c, because when x = 0, y = c. The x-intercepts are the real solutions of ax² + bx + c = 0, if any exist. These intercepts give useful points for sketching the graph.

y 截距总是 c，因为当 x = 0 时，y = c。x 截距是方程 ax² + bx + c = 0 的实数解（如果存在）。这些截距为画图像提供了有用的点。

3. Solving by Factorising | 因式分解法

When a quadratic expression can be factorised into two linear brackets, we can solve the equation by setting each bracket equal to zero. This method uses the zero product property: if pq = 0, then p = 0 or q = 0, or both.

当二次式可以因式分解成两个一次括号时，我们可以令每个括号等于零来解方程。这种方法使用零积性质：如果 pq = 0，那么 p = 0 或 q = 0，或两者都为零。

For example, solve x² − 5x + 6 = 0. Factorising gives (x − 2)(x − 3) = 0, so x − 2 = 0 or x − 3 = 0. Therefore x = 2 or x = 3.

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

x² + bx + c = (x + p)(x + q), where p + q = b and pq = c

Always expand the brackets back out to check that your factorisation is correct. This quick check can prevent sign errors and lost marks.

一定要把括号展开来检查因式分解是否正确。这种快速检查可以防止符号错误和失分。

4. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x − h)² + k. This form is extremely useful because it immediately shows the turning point (h, k) and can help solve equations that do not factorise neatly.

配方法将 ax² + bx + c 改写为 a(x − h)² + k 的形式。这种形式非常有用，因为它能立即显示顶点 (h, k)，并且可以帮助求解不易因式分解的方程。

For example, solve x² + 6x + 5 = 0. Write x² + 6x = −5. Completing the square gives (x + 3)² − 9 = −5, so (x + 3)² = 4. Then x + 3 = ±2, giving x = −1 or x = −5.

例如，解 x² + 6x + 5 = 0。写成 x² + 6x = −5。配方后得到 (x + 3)² − 9 = −5，所以 (x + 3)² = 4。然后 x + 3 = ±2，得到 x = −1 或 x = −5。

x² + bx = (x + b ÷ 2)² − (b ÷ 2)²

When a is not 1, factor out a from the first two terms before completing the square. This step keeps the process accurate for equations such as 2x² + 8x + 5 = 0.

当 a 不等于 1 时，先从前两项中提取公因子 a，再配方。这个步骤能保证像 2x² + 8x + 5 = 0 这样的方程运算准确。

5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is especially useful when factorising is difficult or when the coefficients are decimals or fractions. You can use it in both calculator and non-calculator papers.

求根公式可以解任何二次方程 ax² + bx + c = 0。当因式分解困难或系数为小数、分数时，它特别有用。你可以在计算器和非计算器试卷中使用它。

x = (−b ± √(b² − 4ac)) ÷ 2a

The symbol ± means there are usually two solutions: one using the plus sign and one using the minus sign. If the discriminant is zero, both signs give the same repeated root.

符号 ± 表示通常有两个解：一个使用加号，一个使用减号。如果判别式为零，两个符号会得到相同的重根。

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

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

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

The discriminant is the expression b² − 4ac under the square root in the quadratic formula. It tells you how many real solutions the equation has without needing to solve it completely.

判别式是求根公式中平方根下的表达式 b² − 4ac。它无需完全解方程就能告诉你方程有多少个实数解。

Δ = b² − 4ac

If Δ > 0, the equation has two distinct real roots. If Δ = 0, it has one repeated real root. If Δ < 0, it has no real roots, and the graph does not cross the x-axis.

如果 Δ > 0，方程有两个不同的实数根。如果 Δ = 0，它有一个重根。如果 Δ < 0，它没有实数根，图像不与 x 轴相交。

For example, for x² − 4x + 4 = 0, Δ = 16 − 16 = 0. Therefore the equation has one repeated real root, x = 2.

例如，对于 x² − 4x + 4 = 0，Δ = 16 − 16 = 0。因此该方程有一个重根 x = 2。

7. Graphs: Turning Point and Axis of Symmetry | 图像：顶点与对称轴

The turning point of a parabola is its maximum or minimum point. In completed-square form y = a(x − h)² + k, the turning point is (h, k). The axis of symmetry is the vertical line x = h.

抛物线的顶点是其最大值或最小值点。在配方法形式 y = a(x − h)² + k 中，顶点为 (h, k)。对称轴是竖直线 x = h。

If the quadratic is in standard form, the x-coordinate of the turning point is x = −b ÷ 2a. Substitute this value back into the equation to find the y-coordinate.

如果二次式是一般形式，顶点的 x 坐标是 x = −b ÷ 2a。将该值代回方程即可求出 y 坐标。

x = −b ÷ 2a

For y = x² − 4x + 5, the turning point has x-coordinate x = 4 ÷ 2 = 2. Substituting x = 2 gives y = 1, so the turning point is (2, 1).

对于 y = x² − 4x + 5，顶点 x 坐标为 x = 4 ÷ 2 = 2。代入 x = 2 得 y = 1，因此顶点为 (2, 1)。

8. Sketching Quadratic Graphs | 画二次函数图像草图

To sketch a quadratic graph, find the y-intercept, the x-intercepts if possible, the turning point and the axis of symmetry. Then draw a smooth U-shaped or inverted U-shaped curve through these points.

画二次函数图像草图时，先求 y 截距、如有可能求 x 截距、顶点和对称轴。然后通过这些点画一条平滑的 U 形或倒 U 形曲线。

For y = x² − 4x + 3, the y-intercept is 3. The x-intercepts are x = 1 and x = 3. The turning point is (2, −1). Plot these points and draw a symmetric upward-opening curve.

对于 y = x² − 4x + 3，y 截距为 3。x 截距为 x = 1 和 x = 3。顶点为 (2, −1)。描出这些点，画一条对称的开口向上的曲线。

Always check the sign of a first. If a > 0 the curve opens upward with a minimum; if a < 0 it opens downward with a maximum. This determines the basic shape before you plot any coordinates.

一定要先检查 a 的符号。如果 a > 0，曲线开口向上且有最小值；如果 a < 0，则开口向下且有最大值。这决定了基本形状，然后再描点。

9. Applications and Word Problems | 应用与文字题

Quadratic equations often appear in problems involving area, projectile motion, profit, and number relationships. Translate the words into an equation, solve it, and then check which solutions make sense in the original context.

二次方程经常出现在涉及面积、抛体运动、利润和数字关系的问题中。将文字转化为方程，解出后检查哪些解在原情境中合理。

For example, the product of two consecutive positive integers is 72. Let the smaller integer be x, so x(x + 1) = 72. This gives x² + x − 72 = 0. Factorising gives (x + 9)(x − 8) = 0, so x = 8 because x must be positive.

例如，两个连续正整数的积是 72。设较小的整数为 x，则 x(x + 1) = 72。得到 x² + x − 72 = 0。因式分解得 (x + 9)(x − 8) = 0，所以 x = 8，因为 x 必须为正。

Always state the final answer in the units or form requested, such as cm, m², seconds, or a pair of numbers. This is a common exam requirement and often earns method and accuracy marks.

始终以题目要求的单位或形式陈述最终答案，例如 cm、m²、秒或一组数。这是常见的考试要求，也经常获得方法分和准确分。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

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Quadratic Equations and Graphs | 二次方程与图像