📚 Quadratic Equations: Factorising, Formula and Graphs | 二次方程：因式分解、公式法与图像
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, and problem-solving questions. You need to recognise its standard form, solve it by factorising, by the quadratic formula, or by completing the square, and understand how its graph behaves.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像和应用题中。你需要识别它的标准形式，掌握因式分解法、二次公式法和配方法，并理解其图像的性质。
1. What is a Quadratic Equation? | 什么是二次方程？
known variable, usually x, is 2. It can often be written as ax² + bx + c = 0, where a ≠ 0.
二次方程是一个二次多项式方程。这意味着未知数（通常是 x）的最高次数是 2。它通常可以写成 ax² + bx + c = 0，其中 a ≠ 0。
es linear (bx + c = 0), so the condition a ≠ 0 is essential for a true quadratic equation.
如果 a = 0，方程就变成了线性方程（bx + c = 0），因此条件 a ≠ 0 对于真正的二次方程至关重要。
2. Standard Form and Coefficients | 标准形式与系数
called coefficients: a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.
二次方程的标准形式是 ax² + bx + c = 0。这里 a、b 和 c 被称为系数：a 是二次项系数，b 是一次项系数，c 是常数项。
ation into standard form. For example, 3x² = 5x - 2 should be written as 3x² - 5x + 2 = 0.
在求解之前，一定要先把方程整理成标准形式。例如，3x² = 5x - 2 应该写成 3x² - 5x + 2 = 0。
- Example: In 2x² + 7x - 3 = 0, we have a = 2, b = 7, c = -3.
- 示例：在 2x² + 7x - 3 = 0 中，a = 2，b = 7，c = -3。
3. Solving by Factorisation | 因式分解法
When a quadratic expression can be factorised into two linear brackets, we can solve the equation quickly. The key idea is: if the product of two factors is zero, then at least one factor must be zero.
当一个二次式可以分解为两个一次因式时，我们可以快速求解方程。核心思想是：如果两个因式的乘积为零，那么至少有一个因式为零。
s that multiply to +6 and add to -5. These are -2 and -3, so x² - 5x + 6 = (x - 2)(x - 3).
例如，求解 x² - 5x + 6 = 0。我们寻找两个数，它们相乘得到 +6，相加得到 -5。这两个数是 -2 和 -3，所以 x² - 5x + 6 = (x - 2)(x - 3)。
Then set each bracket equal to zero: x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.
然后让每个括号等于零：x - 2 = 0 或 x - 3 = 0，得到 x = 2 或 x = 3。
When a ≠ 1, use the ac method or trial and error. For 2x² + 7x + 3, the factors are (2x + 1)(x + 3).
当 a ≠ 1 时，可以使用 ac 方法或尝试法。对于 2x² + 7x + 3，因式为 (2x + 1)(x + 3)。
4. The Quadratic Formula | 二次公式
If a quadratic equation cannot be factorised easily, we can use the quadratic formula. It works for every quadratic equation.
如果二次方程不容易因式分解，我们可以使用二次公式。它适用于所有二次方程。
x = (-b ± √b² - 4ac) / 2a
f ax² + bx + c = 0. Pay attention to signs when substituting negative values of a, b or c.
该公式给出 ax² + bx + c = 0 的解。代入 a、b 或 c 的负值时要注意符号。
- 5 = 0. Here a = 2, b = 3, c = -5. Then x = (-3 ± √(3² - 4 × 2 × (-5))) / (2 × 2) = (-3 ± √9 + 40) / 4 = (-3 ± √49) / 4 = (-3 ± 7) / 4.
示例：求解 2x² + 3x - 5 = 0。这里 a = 2，b = 3，c = -5。那么 x = (-3 ± √(3² - 4 × 2 × (-5))) / (2 × 2) = (-3 ± √9 + 40) / 4 = (-3 ± √49) / 4 = (-3 ± 7) / 4。
So x = (-3 + 7)/4 = 1, or x = (-3 - 7)/4 = -10/4 = -2.5.
所以 x = (-3 + 7)/4 = 1，或 x = (-3 - 7)/4 = -10/4 = -2.5。
5. Completing the Square | 配方法
Completing the square is another method that also helps find the turning point of a quadratic graph. The idea is to rewrite ax² + bx + c in the form a(x + p)² + q.
配方法是另一种方法，也有助于求二次函数图像的转折点。其思想是把 ax² + bx + c 改写成 a(x + p)² + q 的形式。
take half of the coefficient of x, which is 3, and write (x + 3)² - 9 + 5 = (x + 3)² - 4.
对于 x² + 6x + 5，我们取 x 的系数的一半，即 3，写成 (x + 3)² - 9 + 5 = (x + 3)² - 4。
solve (x + 3)² - 4 = 0, move 4 across: (x + 3)² = 4, then x + 3 = ±2, so x = -1 or x = -5.
求解 (x + 3)² - 4 = 0，把 4 移过去：(x + 3)² = 4，然后 x + 3 = ±2，所以 x = -1 或 x = -5。
6. Discriminant and Nature of Roots | 判别式与根的性质
c formula: D = b² - 4ac. It tells us the nature of the roots without solving the equation.
判别式是二次公式中平方根里面的表达式：D = b² - 4ac。它能在不求解方程的情况下告诉我们根的性质。
- If D > 0, there are two distinct real roots.
- 如果 D > 0，有两个不相等的实根。
- If D = 0, there is one repeated real root (or two equal roots).
- 如果 D = 0，有一个重实根（或两个相等的根）。
- If D < 0, there are no real roots.
- 如果 D < 0，没有实根。
² - 4x + 4 = 0, D = (-4)² - 4 × 1 × 4 = 16 - 16 = 0, so there is one repeated root, x = 2.
示例：对于 x² - 4x + 4 = 0，D = (-4)² - 4 × 1 × 4 = 16 - 16 = 0，所以有一个重根，x = 2。
7. Graphs of Quadratic Functions | 二次函数图像
. If a > 0, the parabola opens upward and has a minimum point. If a < 0, it opens downward and has a maximum 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，有一个最小值点。如果 a < 0，抛物线开口向下，有一个最大值点。
the constant term c. The x-intercepts are the real roots of the equation ax² + bx + c = 0.
y 轴截距是常数项 c。x 轴截距是方程 ax² + bx + c = 0 的实根。
To sketch the graph, find the y-intercept, the roots (if any), and the turning point.
要画草图，需要找到 y 轴截距、根（如果有的话）和转折点。
8. Turning Point and Axis of Symmetry | 转折点与对称轴
is also called its vertex. The x-coordinate of the turning point is given by x = -b / 2a.
抛物线的转折点也称为顶点。转折点的 x 坐标由 x = -b / 2a 给出。
equation to find the y-coordinate. The vertical line x = -b / 2a is the axis of symmetry.
将这个 x 值代入原方程可求出 y 坐标。竖直直线 x = -b / 2a 就是对称轴。
x = -(-4)/(2 × 1) = 2. Substitute x = 2: y = 4 - 8 + 3 = -1. The turning point is (2, -1).
示例：对于 y = x² - 4x + 3，对称轴是 x = -(-4)/(2 × 1) = 2。代入 x = 2：y = 4 - 8 + 3 = -1。转折点是 (2, -1)。
dratic is written in completed-square form y = (x + p)² + q, the turning point is (-p, q).
如果二次函数写成配方法形式 y = (x + p)² + q，则转折点为 (-p, q)。
9. Quadratic Inequalities | 二次不等式
such as x² - 5x + 6 > 0, first factorise: (x - 2)(x - 3) > 0. The critical values are x =
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