📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from simple factorising to interpreting curved graphs. Knowing how to solve a quadratic equation, how to analyse its discriminant, and how to link the algebra to a parabola is essential for IGCSE success, especially on the Extended tier. This revision guide walks through the standard form, all three solving methods, the discriminant, the shape and key features of quadratic graphs, quadratic inequalities, and the exam technique that turns method marks into full marks.
二次方程贯穿整个 IGCSE 数学大纲，从简单的因式分解到曲线图像的分析。掌握如何求解二次方程、如何分析判别式，以及如何把代数式与抛物线联系起来，是 IGCSE 取得好成绩的关键，尤其在 Extended（拓展）级别。本篇复习指南将依次讲解标准形式、三种求解方法、判别式、二次函数图像的形状与关键特征、二次不等式，以及能把方法分变成满分的考试技巧。
1. What Is a Quadratic Equation? | 什么是二次方程？
re real numbers and a is not equal to zero. The highest power of the unknown is 2, which is exactly what makes the equation quadratic rather than linear. If a were zero, the x² term would disappear and the equation would become linear, so the condition a ≠ 0 is part of the definition, not an optional extra.
二次方程是指任何可以整理成标准形式 a x² + b x + c = 0 的方程，其中 a、b、c 为实数，且 a 不等于零。未知数的最高次数是 2，正是这一点使方程成为二次方程而不是一次方程。如果 a 为零，x² 项就会消失，方程退化为一次方程，因此 a ≠ 0 是定义的一部分，而不是可有可无的附加条件。
a x² + b x + c = 0, a ≠ 0
In IGCSE Mathematics, a quadratic equation is also called an equation of degree two, and it can have two solutions, one solution, or no real solutions. Those solutions are called roots, and they are the values of x that make the expression equal to zero. A quadratic equation never has more than two real roots, because the highest power is 2.
在 IGCSE 数学中，二次方程也被称为二次（次数为二）的方程，它可能有两个解、一个解，或者没有实数解。这些解称为"根"，就是使整个式子等于零的 x 的取值。因为最高次数是 2，所以一个二次方程最多只有两个实数根。
2. Standard Form and Key Terms | 标准形式与各项名称
The term a x² is called the quadratic term, b x is the linear term, and c is the constant term. Recognising these three parts helps you choose the correct solving method and also helps you avoid sign errors. The numbers a, b and c are called coefficients, and they must be read together with their signs.
a x² 项称为二次项，b x 项称为一次项，c 称为常数项。认清这三部分有助于选择正确的求解方法，也能帮助你避免符号错误。a、b、c 这三个数称为系数，读的时候必须带上它们各自的符号。
re you factorise or apply the formula. Here a = 3, b = −5 and c = 2. Working with a non-zero right-hand side is the single most common cause of lost marks in quadratic questions.
在动手之前，务必先把方程整理成一边等于零的形式。例如 3x² = 5x − 2 必须先改写为 3x² − 5x + 2 = 0，才能进行因式分解或代入公式。此时 a = 3，b = −5，c = 2。右边不为零就动手计算，是二次方程题目中最常见的失分原因。
3x² − 5x + 2 = 0: a = 3, b = −5, c = 2
− 3x − 4 = 6 and then to x² − 3x − 10 = 0. Similarly, equations involving fractions should be multiplied through by the common denominator first.
有些方程会隐藏自己的二次身份。像 (x + 1)(x − 4) = 6 这样的方程看上去已经分解好了，但其实还不能直接求解：必须先展开为 x² − 3x − 4 = 6，再整理为 x² − 3x − 10 = 0。含有分母的方程同样应先乘以公分母，去掉分数后再整理。
3. Solving by Factorising | 因式分解法求解
umbers is zero, then at least one of them must be zero. So if (x − p)(x − q) = 0, then x − p = 0 or x − q = 0, giving x = p or x = q.
当方程的根是整数或简单分数时，因式分解通常是最快的方法。它的依据是"零积法则"：如果两个数的乘积为零，那么至少有一个数为零。因此若 (x − p)(x − q) = 0，则 x − p = 0 或 x − q = 0，即 x = p 或 x = q。
− 5x + 6 = 0, find two numbers that multiply to 6 and add to −5: they are −2 and −3. So (x − 2)(x − 3) = 0 and the roots are x = 2 and x = 3.
对于二次项系数为 1 的方程 x² + b x + c，要找两个数，使它们的积为 c、和为 b，于是 x² + (p + q)x + p q = (x + p)(x + q)。例如解 x² − 5x + 6 = 0，需要两个数积为 6、和为 −5，即 −2 和 −3，所以 (x − 2)(x − 3) = 0，根为 x = 2 和 x = 3。
x² − 5x + 6 = (x − 2)(x − 3) = 0 → x = 2 or x = 3
m b = 7 give 6 and 1, so the middle term splits as 2x² + 6x + x + 3 = 0, which factorises to (2x + 1)(x + 3) = 0, giving x = −1/2 or x = −3.
当 x² 的系数不是 1 时，可以对 a c（而不是 c）使用积和法，或者拆分中间项。对于 2x² + 7x + 3 = 0，乘积 a c = 6、和 b = 7，得到 6 和 1，于是中间项拆成 2x² + 6x + x + 3 = 0，分解为 (2x + 1)(x + 3) = 0，因此 x = −1/2 或 x = −3。
nce of two squares: x² − 9 = 0 becomes (x + 3)(x − 3) = 0, giving x = ±3. Never divide both sides by x, because that destroys the root x = 0.
有两种特殊情形能节省大量时间。第一，先提取公因式：3x² − 12x = 0 变成 3x(x − 4) = 0，得 x = 0 或 x = 4。第二，使用平方差公式：x² − 9 = 0 变成 (x + 3)(x − 3) = 0，得 x = ±3。千万不要两边同时除以 x，那会丢掉 x = 0 这个根。
4. Solving by Completing the Square | 配方法求解
+ p)² + q in general. It always works, it gives exact surd answers, and it reveals the vertex of the parabola at the same time. The rule for the monic case is simple: halve the coefficient of x, put it inside the bracket, and subtract its square.
配方法把二次式改写成 (x + p)² + q = 0 的形式，一般情况下写成 a(x + p)² + q。它总是可行，能给出精确的根式答案，同时还能直接读出抛物线的顶点。二次项系数为 1 时的规则很简单：取一次项系数的一半放进括号，再减去它的平方。
x² + b x + c = (x + b/2)² − (b/2)² + c
1 = (x + 3)² − 9 + 1 = (x + 3)² − 8. Setting this to zero gives (x + 3)² = 8, so x + 3 = ±√8, and therefore x = −3 ± 2√2. Rounded to three significant figures, x = −0.172 and x = −5.83.
示例：解 x² + 6x + 1 = 0。6 的一半是 3，于是 x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8。令其等于零，得 (x + 3)² = 8，所以 x + 3 = ±√8，即 x = −3 ± 2√2。保留三位有效数字，x = −0.172 与 x = −5.83。
5 = 0, write 2(x² − 4x) + 5 = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3. Solving gives (x − 2)² = 3/2, so x = 2 ± √3/2 ≈ 2 ± 1.22, that is x ≈ 3.22 or x ≈ 0.775.
当 x² 的系数不是 1 时，只把该系数从前两项中提出，在括号内配方，再乘回去。对于 2x² − 8x + 5 = 0，写成 2(x² − 4x) + 5 = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 3。求解得 (x − 2)² = 3/2，所以 x = 2 ± √3/2 ≈ 2 ± 1.22，即 x ≈ 3.22 或 x ≈ 0.775。
The completed-square form is also the quickest way to answer questions such as "find the minimum value of x² + 6x + 1" or "show that x² + 6x + 1 is always greater than −8". The minimum value is the constant q at the end, occurring when the bracket equals zero.
配方式也是回答"求 x² + 6x + 1 的最小值"或"证明 x² + 6x + 1 恒大于 −8"这类问题的最快方法。最小值就是最后的常数 q，它在括号内等于零时取得。
5. Solving with the Quadratic Formula | 用求根公式求解
dard form, and it is the method of choice when factorising fails or when the question says "give your answers correct to three significant figures". It comes directly from completing the square on the general equation a x² + b x + c = 0.
求根公式可以求解任何已经整理成标准形式的二次方程，当因式分解行不通，或题目要求"答案保留三位有效数字"时，它是首选方法。它正是对一般方程 a x² + b x + c = 0 配方后得到的直接结果。
x = ( −b ± √b² − 4ac ) ÷ 2a
± 5) ÷ 4. This gives x = 3 or x = 0.5. Notice that the answer is exact, and the fact that √25 s a whole number tells you the equation could have been factorised as (2x − 1)(x − 3) = 0.
示例：解 2x² − 7x + 3 = 0。此处 a = 2，b = −7，c = 3，所以 b² − 4ac = 49 − 24 = 25，x = (7 ± 5) ÷ 4，得 x = 3 或 x = 0.5。注意答案是精确值；而 √25 是整数这一事实也告诉你，该方程本可以分解为 (2x − 1)(x − 3) = 0。
When using the formula in an exam, always write down the values of a, b and c before substituting. Then substitute in one clean step, keeping the bracket around the negative b and around the whole discriminant. Common slips include forgetting the minus sign in −b, using 2a instead of 2 multiplied by a, and rounding the square root too early, which can shift the final answer beyond the accepted range.
考试中使用公式时，先把 a、b、c 的值写下来，再代换。代换时保持 −b 的括号和整个判别式的括号，一步到位。常见失误包括漏掉 −b 的负号、把 2a 算错，以及过早对根号取近似值，导致最终答案超出可接受范围。
6. The Discriminant and the Number of Roots | 判别式与根的个数
The expression under the square root, b² − 4ac, is called the discriminant and is often written as Δ. It tells you how many real solutions a quadratic equation has, without solving it. This is a standard Extended tier topic and appears in both algebra and graph questions.
根号下的表达式 b² − 4ac 称为判别式，常记作 Δ。它能让你在不求解的情况下判断二次方程有几个实数解。这是 Extended 级别的标准考点，在代数题和图像题中都会出现。
Δ = b² − 4ac
| Value of Δ | 判别式的值 | Number of real roots | 实根个数 | Graph meaning | 图像意义 |
| Δ > 0 | Two distinct real roots | 两个不同实根 | Curve cuts the x-axis twice | 曲线与 x 轴相交两次 |
| Δ = 0 | One repeated root | 一个重根 | Curve touches the x-axis | 曲线与 x 轴相切 |
| Δ < 0 | No real roots | 无实根 | Curve misses the x-axis | 曲线与 x 轴不相交 |
no real roots. Here Δ = 16 − 28 = −12, which is negative, so there are no real solutions.
例题：求使 x² + k x + 9 = 0 有重根的 k 值。重根意味着 Δ = 0，所以 k² − 36 = 0，k = ±6。例题：证明 x² + 4x + 7 = 0 无实根。此处 Δ = 16 − 28 = −12，为负数，所以没有实数解。
a perfect square, the roots are irrational surds, which is why the formula leaves them in the form p ± q√r.
如果 a、b、c 都是有理数且 Δ 是完全平方数，则根为有理数，二次式可在整数范围内分解。如果 Δ 为正但不是完全平方数，根就是无理根式，这正是公式把答案保留成 p ± q√r 形式的原因。
7. Graphs of Quadratic Functions: The Parabola | 二次函数图像：抛物线
The graph of y = a x² + b x + c is a parabola — is the bridge between the algebra of the previous sections and the geometry of the graph.
y = a x² + b x + c 的图像是一条抛物线——一条光滑、对称的 U 形曲线。a 的符号决定开口方向，a 的大小决定曲线的宽窄。这正是前面代数内容与图像几何之间的桥梁。
um point; if a < 0, it opens downwards and has a maximum point. The larger the value of |a|, the narrower the parabola; the smaller |a| becomes. The constant c is always the y-intercept, because substituting x = 0 gives y = c.
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。|a| 越大，抛物线越窄；|a| 越小，曲线越宽越平缓。常数 c 永远是 y 轴截距，因为令 x = 0 得 y = c。
y = a x² + b x + c: a > 0 opens up, a < 0 opens down
; y = −f(x) reflects it in the x-axis; and y = f(k x) stretches or squeezes it horizontally. For parabolas, these transformations move the vertex and change the number of roots, so always re-check the x-intercepts after a transformation.
IGCSE 还会考查简单的图像变换。y = f(x) + k 是把 y = f(x) 向上平移 k 个单位；y = f(x + k) 是向左平移 k 个单位；y = −f(x) 是关于 x 轴翻折；y = f(k x) 是水平方向的拉伸或压缩。对于抛物线，这些变换会移动顶点并改变根的个数，因此变换之后一定要重新检查与 x 轴的交点。
8. Axis of Symmetry, Vertex and Turning Point | 对称轴、顶点与极值点
This value is also the x-coordinate of the vertex, which is the turning point of the curve. Because the curve is symmetric, the vertex lies exactly halfway between the two roots.
每条抛物线都有一条竖直的对称轴，其方程是 x = −b ÷ 2a。这个值同时也是顶点（即曲线的极值点）的横坐标。由于曲线对称，顶点恰好位于两个根的正中间。
Axis of symmetry: x = −b ÷ 2a
mum point because a = 1 > 0. The completed-square form confirms this: y = (x − 2)² − 3.
示例：对于 y = x² − 4x + 1，对称轴为 x = 4 ÷ 2 = 2。代入 x = 2 得 y = 4 − 8 + 1 = −3，所以顶点是 (2, −3)，由于 a = 1 > 0，它是最低点。配方式也印证了这一点：y = (x − 2)² − 3。
in factorised form y = a(x − p)(x − q), the axis of symmetry is the midpoint x = (p + q) ÷ 2. Choosing the right form saves time and reduces the chance of arithmetic mistakes.
如果方程以配方形式 y = a(x + p)² + q 给出，可以直接读出顶点为 (−p, q)，无需计算。如果以分解形式 y = a(x − p)(x − q) 给出，对称轴就是中点 x = (p + q) ÷ 2。选择合适的形式既省时，又能减少算错的机会。
9. Roots, Intercepts and the Graph | 根、截距与图像
-axis. This single idea links the algebra and the graph, and it explains why the discriminant works the way it does.
方程 a x² + b x + c = 0 的根，正是图像 y = a x² + b x + c 与 x 轴交点的横坐标。这一条思想把代数与图像联系起来，也解释了判别式为什么那样起作用。
crosses the x-axis at (−2, 0) and (3, 0). The axis of symmetry is x = 0.5, and substituting gives y = −6.25, so the vertex is (0.5, −6.25). Because a = 1 > 0 the curve opens upwards.
示例：画出 y = x² − x − 6 的图像。y 轴截距为 c = −6。分解得 (x − 3)(x + 2) = 0，所以根为 x = 3 与 x = −2，曲线与 x 轴交于 (−2, 0) 和 (3, 0)。对称轴为 x = 0.5，代入得 y = −6.25，故顶点为 (0.5, −6.25)。因为 a = 1 > 0，曲线开口向上。
y = x² − x − 6: roots x = −2, 3; vertex (0.5, −6.25)
Substituting the point gives 8 = k(−1)(−4) = 4k, so k = 2 and the equation is y = 2x² − 10x + 8.
反过来，也可以从图像读出信息。若图像与 x 轴交于 x = 1 和 x = 4，且经过点 (0, 8)，则方程必为 y = k(x − 1)(x − 4) 的形式。代入该点得 8 = k(−1)(−4) = 4k，所以 k = 2，方程为 y =
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