📚 Mastering Quadratic Equations and Graphs for IGCSE Mathematics | IGCSE 数学：二次方程与图像精讲
In IGCSE Mathematics, quadratic equations and graphs appear in both Paper 1 and Paper 2, connecting algebra, coordinate geometry and real-world problem solving. This revision guide walks through expanding double brackets, factorising, solving by factorisation, completing the square, using the quadratic formula, and interpreting parabolic graphs.
在 IGCSE 数学中，二次方程与图像在试卷一和试卷二中均有出现，它们把代数、坐标几何与现实问题求解联系起来。本复习指南将系统讲解双括号展开、因式分解、因式分解法解方程、配方法、求根公式以及抛物线图像解读。
1. What Is a Quadratic Expression? | 什么是二次表达式？
A quadratic expression is any expression of the form ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which is why it is called quadratic.
二次表达式是形如 ax² + bx + c 的表达式，其中 a、b、c 为常数且 a ≠ 0。x 的最高次数为 2，因此称为二次。
expressions. If a = 0, the expression becomes linear, so the condition a ≠ 0 is essential.
例如，3x² − 5x + 2、x² + 4x 和 −2x² + 7 都是二次表达式。如果 a = 0，表达式就变成一次式，因此 a ≠ 0 这一条件必不可少。
2. Expanding Double Brackets | 双括号展开
Expanding two brackets is often the first step in working with quadratics. For (x + p)(x + q), multiply each term in the first bracket by each term in the second bracket.
展开双括号通常是处理二次式的第一步。对于 (x + p)(x + q)，需要把第一个括号中的每一项分别乘以第二个括号中的每一项。
(x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15
t is not 1, the same principle applies: (2x + 1)(x − 4) = 2x² − 8x + x − 4 = 2x² − 7x − 4.
中间两项合并后得到 x 的系数。当首项系数不为 1 时，同样原理也适用：(2x + 1)(x − 4) = 2x² − 8x + x − 4 = 2x² − 7x − 4。
3. Factorising Quadratics of the Form x² + bx + c | 形如 x² + bx + c 的因式分解
mbers p and q such that p + q = b and pq = c. Then write the expression as (x + p)(x + q).
要对 x² + bx + c 进行因式分解，需要找到两个数 p 和 q，使 p + q = b 且 pq = c。然后把表达式写成 (x + p)(x + q)。
x² + 7x + 10 = (x + 2)(x + 5)
signs, check carefully: x² − 3x − 18 = (x − 6)(x + 3), since −6 + 3 = −3 and −6 × 3 = −18.
这里 2 + 5 = 7 且 2 × 5 = 10。含有负号时要仔细检查：x² − 3x − 18 = (x − 6)(x + 3)，因为 −6 + 3 = −3 且 −6 × 3 = −18。
4. Factorising Quadratics of the Form ax² + bx + c | 形如 ax² + bx + c 的因式分解
When a > 1, factorising is slightly harder. One reliable method is to split the middle term using two numbers whose product is ac and whose sum is b.
当 a > 1 时，因式分解会稍难一些。一种可靠的方法是使用两个数来拆分中间项，这两个数的乘积为 ac，和为 b。
2x² + 7x + 3: ac = 2 × 3 = 6, so use 6 and 1
rs: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Always expand at the end to check your answer.
重写中间项：2x² + 6x + x + 3。分组提取公因式：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。最后一定要展开检验答案。
5. Solving Quadratic Equations by Factorising | 因式分解法解二次方程
To solve a quadratic equation, first rearrange it so that one side is 0. Factorise the quadratic expression, then use the zero product property: if a product equals zero, at least one factor must be zero.
解二次方程时，先把方程化为一边为 0 的形式。对二次表达式进行因式分解，然后利用零乘积性质：如果乘积为 0，那么至少有一个因式为 0。
x² − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3
Do not divide both sides by x unless you know x ≠ 0, because that can remove a valid solution. Always set each factor equal to zero separately.
除非你确定 x ≠ 0，否则不要在方程两边同时除以 x，因为这可能会丢掉一个有效解。应始终分别令每个因式等于零。
6. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x − h)² + k. This form immediately reveals the vertex of the parabola and helps solve equations.
配方法把 ax² + bx + c 改写为 a(x − h)² + k 的形式。这种形式可以立即揭示抛物线的顶点，也有助于解方程。
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
rom the x terms: 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3.
对于 2x² + 8x + 5，先从含 x 的项中提出 2：2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3。
7. The Quadratic Formula and the Discriminant | 求根公式与判别式
uadratic equation ax² + bx + c = 0. It is especially useful when factorising is difficult.
求根公式可以解任何二次方程 ax² + bx + c = 0。当因式分解较困难时，它尤其有用。
x = (−b ± √b² − 4ac) ÷ 2a
The expression b² − 4ac is called the discriminant, often written as Δ. It tells you how many real roots the equation has without solving it fully.
表达式 b² − 4ac 称为判别式，通常写作 Δ。它可以在不完全解方程的情况下告诉你方程有多少个实数根。
| Discriminant Δ = b² − 4ac | Number of real roots | Graph intersects x-axis |
|---|---|---|
| Δ > 0 | 2 distinct real roots | two points |
| Δ = 0 | 1 repeated real root | one point, the x-axis is a tangent |
| Δ < 0 | no real roots | does not cross the x-axis |
ten as ax² + bx + c = 0. Substitute the values carefully and use brackets on a calculator.
请记住，只有把方程先写成 ax² + bx + c = 0 后，公式才能给出根。代入数值时要仔细，并在计算器中使用括号。
8. Graphs of Quadratic Functions | 二次函数图像
a. If a > 0, it opens upward like a smile; if a < 0, it opens downward like a frown.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，它开口向上，形似微笑；如果 a < 0，它开口向下，形似皱眉。
. The axis of symmetry is the vertical line x = −b ÷ 2a, and the vertex lies on this line.
常数 c 是 y 轴截距，因为当 x = 0 时 y = c。对称轴是竖直线 x = −b ÷ 2a，顶点位于这条线上。
Axis of symmetry: x = −b ÷ 2a
9. Interpreting Quadratic Graphs: Roots, Vertex, Axis of Symmetry | 解读二次图像：根、顶点、对称轴
The roots of a quadratic equation are the x-coordinates where the graph crosses the x-axis. The vertex is the maximum or minimum point of the parabola.
二次方程的根是图像与 x 轴交点的 x 坐标。顶点是抛物线的最高点或最低点。
ly (h, k). For example, y = 2(x − 3)² − 10 has vertex (3, −10) and axis of symmetry x = 3.
如果二次函数写成完全平方形式 y = a(x − h)² + k，那么顶点就是 (h, k)。例如，y = 2(x − 3)² − 10 的顶点为 (3, −10)，对称轴为 x = 3。
If a > 0, the vertex is the minimum point; if a < 0, it is the maximum point. Exam questions often ask for the coordinates of the vertex, so completing the square is a valuable skill.
如果 a > 0，顶点是最低点；如果 a < 0，顶点是最高点。考试题经常要求求顶点坐标，因此配方法是一项很有用的技能。
10. Applications and Word Problems | 应用与文字题
ght of a projectile might be given by h = −5t² + 20t + 1, where h is height and t is time.
许多现实情境都可以用二次函数建模。例如，抛体的高度可以用 h = −5t² + 20t + 1 表示，其中 h 为高度，t 为时间。
To solve word problems, define the unknown clearly, form a quadratic equation, solve it, and then check that the answer makes sense in the original context. Reject negative solutions when they are impossible for lengths, times or counts.
解文字题时，要清楚地设定未知数，列出二次方程，解方程，然后检查答案在原始情境中是否合理。如果长度、时间或数量不可能为负数，则应舍去负解。
Rectangle problem: width = x, length = x + 3, area = 54
54 = 0. Factorise to (x + 9)(x − 6) = 0, so x = 6 is the valid width and the length is 9.
于是 x(x + 3) = 54，得到 x² + 3x − 54 = 0。因式分解为 (x + 9)(x − 6) = 0，所以 x = 6 是合理的宽度，长度为 9。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is forgetting to set the equation to zero before factorising. Another is losing a factor when dividing by a variable, or copying a negative sign incorrectly during expansion.
一个常见错误是因式分解前忘记把方程化为等于 0 的形式。另一个错误是用变量相除时丢掉一个因式，或者在展开时抄错负号。
When using the quadratic formula, do not drop the brackets around negative b values. Write the expression step by step and use the calculator carefully. Always substitute your final roots back into the original equation to check.
使用求根公式时，不要把负 b 值周围的括号丢掉。应逐步写出表达式并仔细使用计算器。最后一定要把得到的根代回原方程进行检验。
For graph sketching, find the y-intercept, roots if factorised, axis of symmetry and vertex. These key points are usually enough to draw a correct parabola and earn full marks.
画图像时，应找出 y 轴截距、根（如果可以因式分解）、对称轴和顶点。这些关键点通常足以画出正确的抛物线并获得满分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图