📚 Solving Quadratic Equations | 解二次方程
A quadratic equation is a polynomial equation of degree two. In IGCSE mathematics, solving these equations is a core skill that appears in algebra, graphs, and word problems. This article reviews all standard methods and common pitfalls.
二次方程是次数为2的多项式方程。在IGCSE数学中，解二次方程是代数、图象和应用题中的核心技能。本文将复习所有标准方法与常见陷阱。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation in one variable can be written in the standard form:
一个变量的二次方程可以写成标准形式：
ax² + bx + c = 0, a ≠ 0
Here a, b, c are real numbers/constants, and a is not zero because otherwise the equation becomes linear. The values of b and c can be zero.
这里a、b、c为实数常数，且a不能为0，否则方程就变成一次方程。b和c可以为0。
Example: 2x² – 3x + 1 = 0 is in standard form with a = 2, b = -3, c = 1.
例如：2x² – 3x + 1 = 0是标准形式，其中a = 2，b = -3，c = 1。
It is important to rearrange any given equation into this standard form before trying to solve it. Many IGCSE questions include an extra simplification step before the quadratic becomes apparent.
在求解前，务必先将给定方程整理成这种标准形式。很多IGCSE题目会先需要化简，才会出现二次方程。
2. Solving by Factorisation | 因式分解法
If the quadratic expression can be factorised, we use the zero product property: if AB = 0 then A = 0 or B = 0.
如果二次表达式可以因式分解，我们利用零积性质：若AB = 0，则A = 0或B = 0。
Steps:
步骤：
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Rearrange the equation so one side is zero.
将方程整理为一边为0。
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Factorise the quadratic expression completely.
将二次表达式完全因式分解。
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Set each factor equal to zero and solve the resulting linear equations.
令每个因式等于0，并解所得一次方程。
Example: Solve x² – 5x + 6 = 0.
例：解方程x² – 5x + 6 = 0。
(x – 2)(x – 3) = 0
x = 2 or x = 3
For leading coefficient a ≠ 1, look for factors of ac that add to b. For example, 2x² – 5x – 3 = 0 can be factorised as (2x + 1)(x – 3) = 0, giving x = -1/2 or x = 3.
当a ≠ 1时，寻找ac的因子使其和为b。例如，2x² – 5x – 3 = 0可分解为(2x + 1)(x – 3) = 0，得x = -1/2或x = 3。
Notice that we must not divide both sides by x if x may be zero; that would lose a solution.
注意：如果x可能为0，不能在方程两边同除以x，否则会丢失一个解。
3. Solving by Taking Square Roots | 平方根法
When the equation has the form (x + p)² = q, we can solve by taking square roots directly.
当方程具有(x + p)² = q的形式时，可以直接两边开平方来求解。
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If q > 0, there are two real solutions: x + p = ±√q.
若q > 0，有两个实数解：x + p = ±√q。
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If q = 0, there is one repeated solution: x = -p.
若q = 0，有一个重根：x = -p。
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If q < 0, there are no real solutions (in IGCSE we do not deal with imaginary numbers).
若q < 0，没有实数解（IGCSE不涉及虚数）。
Example: Solve (x – 3)² = 7.
例：解(x – 3)² = 7。
x – 3 = ±√7 → x = 3 ± √7
This method is particularly useful when the equation is already in completed-square form, or when a quick exact answer is needed without using the quadratic formula.
当方程已经是完全平方式形式，或需要快速精确答案而不想用二次公式时，这种方法特别有用。
4. Completing the Square | 配方法
Completing the square transforms a quadratic into the form a(x + h)² + k. The key identity is:
配方法将二次式变换为a(x + h)² + k的形式。关键恒等式是：
x² + px = (x + p/2)² – (p/2)²
For example, x² + 6x = (x + 3)² – 9. Then we can solve by setting the expression equal to zero and using the square root method.
例如，x² + 6x = (x + 3)² – 9。然后令表达式等于0，再用平方根法求解。
Example: Solve x² + 6x + 4 = 0.
例：解x² + 6x + 4 = 0。
(x + 3)² – 9 + 4 = 0 → (x + 3)² = 5
x = -3 ± √5
When a ≠ 1, first factor out a before completing the square. For example, 2x² + 8x + 3 = 0 becomes 2[(x + 2)² – 4] + 3 = 0, then solve for x.
当a ≠ 1时，先提出a再配方。例如，2x² + 8x + 3 = 0变为2[(x + 2)² – 4] + 3 = 0，再求出x。
5. The Quadratic Formula | 二次公式
For a quadratic equation ax² + bx + c = 0 with a ≠ 0, the solutions are given by the quadratic formula:
对于二次方程ax² + bx + c = 0（a ≠ 0），解由二次公式给出：
x = (-b ± √(b² – 4ac)) / (2a)
This formula works for every quadratic equation, including those that cannot be factorised easily. You should memorise it and be able to use it accurately.
这个公式适用于所有二次方程，包括那些不易因式分解的方程。你应当熟记并能准确使用它。
Example: Solve 2x² + 3x – 2 = 0 using the formula.
例：用公式解2x² + 3x – 2 = 0。
Here a = 2, b = 3, c = -2. Substitute:
这里a = 2，b = 3，c = -2。代入：
x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4
x = 0.5 or x = -2Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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