Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to measurement and modelling. This revision guide explains how to recognise, solve and interpret quadratic equations using factorising, completing the square and the quadratic formula.

二次方程贯穿 IGCSE 数学考纲，从代数、图像到测量与建模都会出现。本复习指南讲解如何识别、求解并解释二次方程，方法包括因式分解、配方法和求根公式。

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

n x is x². The general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是次数为 2 的多项式方程。未知数 x 的最高次幂是 x²。一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。

ecause the x² term disappears. Quadratic equations can have two solutions, one repeated solution, or no real solutions depending on their coefficients.

如果 a = 0，方程就变成一元一次方程，因为 x² 项消失了。二次方程可能有两个解、一个重复解，或者没有实数解，具体取决于系数。

d −x² + 3x = 0. Each can be solved by a suitable method after writing it in standard form.

常见例子包括 x² + 5x + 6 = 0、2x² − 8 = 0 和 −x² + 3x = 0。写成标准形式后，每个方程都可以用合适的方法求解。

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2. Standard Form and Key Terms | 标准形式与关键术语

= 0. This means collecting all terms on one side and setting the other side equal to zero.

求解前，要把方程写成标准形式 ax² + bx + c = 0。也就是说，把所有项移到一边，另一边等于 0。

s the constant term. When a = 1, the quadratic is called monic; otherwise it is non-monic.

系数 a 是 x² 的系数，b 是 x 的系数，c 是常数项。当 a = 1 时，这个二次式称为首一二次式；否则为非首一二次式。

Solutions of a quadratic equation are also called roots, zeros or x-intercepts of the related graph. You should always rearrange terms carefully and avoid moving terms across the equals sign without changing signs.

二次方程的解也叫根、零点或相应图像与 x 轴的交点。移项时一定要仔细，不要在不改变符号的情况下把项移到等号另一边。

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3. Solving by Factorising | 因式分解法

ion can be solved quickly using the zero-product property: if pq = 0, then p = 0 or q = 0.

当二次式可以因式分解时，可以利用零乘积性质快速求解：如果 pq = 0，那么 p = 0 或 q = 0。

= 0. Setting each bracket equal to zero gives x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.

例：求解 x² + 5x + 6 = 0。因式分解得到 (x + 2)(x + 3) = 0。令每个括号等于 0，得到 x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。

1/2 or x = −3. Always check by substituting the solutions back into the original equation.

对于非首一二次式，例如 2x² + 7x + 3 = 0，因式分解得到 (2x + 1)(x + 3) = 0，所以 x = −1/2 或 x = −3。一定要把解代回原方程检验。

• Write the equation with zero on one side
• Factorise the quadratic expression completely
• Set each factor equal to zero
• Solve each linear equation

先将方程化为一边等于 0；再将二次式完全因式分解；然后令每个因式等于 0；最后解每个一元一次方程。

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4. Solving by Completing the Square | 配方法

Completing the square rewrites x² + bx as (x + p)² − q. This method is especially useful for deriving the quadratic formula and for finding the vertex of a parabola.

配方法把 x² + bx 写成 (x + p)² − q。这种方法在推导求根公式和求抛物线顶点时特别有用。

6 is 3, so add 3² = 9 to both sides: x² + 6x + 9 = 4. The left side becomes (x + 3)² = 4.

以 x² + 6x + 5 = 0 为例，首先把 x² + 6x 留在一边：x² + 6x = −5。6 的一半是 3，所以两边加上 3² = 9：x² + 6x + 9 = 4。左边变为 (x + 3)² = 4。

= −1 or x = −5. When the coefficient of x² is not 1, divide the whole equation by a first.

开平方得到 x + 3 = ±2，所以 x = −1 或 x = −5。当 x² 的系数不是 1 时，要先把整个方程除以 a。

ng point at (h, k). This will be explored further in the graphical interpretation section.

配方后，表达式 y = a(x − h)² + k 的转折点在 (h, k)。这个内容将在图像解释一节中进一步讲解。

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5. The Quadratic Formula | 求根公式

tic equation in standard form. It is derived by completing the square on ax² + bx + c = 0.

求根公式可以求解任何标准形式的二次方程。它是由对 ax² + bx + c = 0 进行配方推导出来的。

x = (−b ± √b² − 4ac) ÷ 2a

x = (−b ± √b² − 4ac) ÷ 2a

b² − 4ac, is called the discriminant. The ± symbol means there are usually two solutions.

平方根下的式子 b² − 4ac 称为判别式。± 符号表示通常有两个解。

Example: For 2x² + 3x − 2 = 0, a = 2, b = 3, c = −2. Then x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4, giving x = 1/2 or x = −2.

例：对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2。则 x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4，得到 x = 1/2 或 x = −2。

Always write down a, b and c before substituting into the formula, and simplify the square root carefully. If the discriminant is not a perfect square, leave your answer in surd form unless a decimal is requested.

代入公式前一定要先写下 a、b、c，并仔细化简平方根。如果判别式不是完全平方数，除非题目要求小数，否则答案保留根式形式。

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

nt D = b² − 4ac tells you how many real roots a quadratic equation has without solving it.

判别式 D = b² − 4ac 可以在不求解的情况下判断二次方程有多少个实数根。

Discriminant	Number and type of roots
D > 0	Two distinct real roots
D = 0	One repeated real root
D < 0	No real roots

判断根的情况：D > 0 时有两个不同的实数根；D = 0 时有一个重复的实数根；D < 0 时没有实数根。

This is useful for questions asking whether a graph cuts, touches or avoids the x-axis. A positive discriminant means the parabola cuts the x-axis twice, a zero discriminant means it touches the axis once, and a negative discriminant means it never intersects the x-axis.

这在判断图像是与 x 轴相交、相切还是不相交时很有用。判别式为正意味着抛物线与 x 轴相交两次；判别式为零意味着与 x 轴相切一次；判别式为负意味着不与 x 轴相交。

or x² − 4x + 5 = 0, D = (−4)² − 4(1)(5) = 16 − 20 = −4, so the equation has no real roots.

例：对于 x² − 4x + 5 = 0，D = (−4)² − 4(1)(5) = 16 − 20 = −4，所以该方程没有实数根。

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7. Graphical Interpretation | 图像解释

f y = ax² + bx + c is a parabola. Its x-intercepts are the real roots of ax² + bx + c = 0.

y = ax² + bx + c 的图像是一条抛物线。抛物线与 x 轴的交点就是 ax² + bx + c = 0 的实数根。

If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The coefficient a also affects the width of the parabola.

如果 a > 0，抛物线开口向上，有最低点；如果 a < 0，抛物线开口向下，有最高点。系数 a 还会影响抛物线的宽窄。

are: y = a(x − h)² + k has vertex (h, k). The line of symmetry is the vertical line x = h.

顶点或转折点可以用配方法求得：y = a(x − h)² + k 的顶点是 (h, k)。对称轴是竖直线 x = h。

= (x − 2)² − 1, so the vertex is (2, −1) and the graph cuts the x-axis at x = 1 and x = 3.

对于 y = x² − 4x + 3，配方得到 y = (x − 2)² − 1，所以顶点是 (2, −1)，图像与 x 轴在 x = 1 和 x = 3 处相交。

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