Mastering Quadratic Equations | 精通二次方程

📚 Mastering Quadratic Equations | 精通二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in pure algebra, coordinate geometry, and real-life word problems. A good understanding of the standard form, solving methods, and the discriminant will help you answer many examination questions confidently.

二次方程是IGCSE数学中最重要的主题之一。它们在纯代数、坐标几何以及实际应用题中都会出现。熟悉标准形式、求解方法以及判别式，能帮助你在考试中自信地解决许多问题。

1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is a polynomial equation of degree 2. The general form is

二次方程是最高次数为2的多项式方程。它的一般形式为

ax² + bx + c = 0

where a, b and c are constants, and a is not equal to 0. The constant a is called the quadratic coefficient, b is the linear coefficient, and c is the constant term.

其中a、b和c为常数，且a不等于0。常数a称为二次项系数，b为一次项系数，c为常数项。

If a = 0, the equation becomes bx + c = 0, which is linear. Therefore, the condition a ≠ 0 is essential for the equation to be quadratic.

若a = 0，方程退化为bx + c = 0，即线性方程。因此，a ≠ 0是方程成为二次方程的关键条件。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the roots are rational numbers. We try to rewrite the quadratic expression as the product of two linear brackets. By the zero-product property, if the product is zero then at least one factor must be zero.

当根为有理数时，因式分解通常是最快的方法。我们将二次表达式改写成两个线性括号的乘积。根据零积性质，若乘积为零，则至少有一个因式必须为零。

Step 1: Write the equation in the form x² + bx + c = 0, if possible.

第一步：尽可能将方程写成x² + bx + c = 0的形式。
Step 2: Find two numbers whose product is c and whose sum is b.

第二步：找出两个数，使它们的积为c，和为b。
Step 3: Write as (x + m)(x + n) = 0 and solve each bracket set to zero.

第三步：写成(x + m)(x + n) = 0，并令每个括号分别为零求解。

For example, x² − 5x + 6 = 0 can be factorised as (x − 2)(x − 3) = 0, so x = 2 or x = 3.

例如，x² − 5x + 6 = 0可以分解为(x − 2)(x − 3) = 0，因此x = 2或x = 3。

3. Solving by Taking Square Roots | 平方根法

If the equation is in the form x² = d, we can solve it directly by taking square roots.

如果方程的形式为x² = d，我们可以直接通过开平方来求解。

x = ±√d

When d > 0 there are two real solutions; when d = 0 there is exactly one solution; when d < 0 there are no real solutions.

当d > 0时，有两个实数解；当d = 0时，只有一个解；当d < 0时，没有实数解。

For example, from 4x² = 36 we get x² = 9, so x = ±3.

例如，由4x² = 36可得x² = 9，因此x = ±3。

This method is especially useful for equations of the form (x + h)² = k, which often arise after completing the square.

这种方法特别适用于形如(x + h)² = k的方程，而这种形式经常在配方后出现。

4. Completing the Square | 配方法

Completing the square converts ax² + bx + c into a perfect square plus a constant. This method is useful for solving equations and for identifying the turning point of a parabola.

配方法将ax² + bx + c化为完全平方加常数项。该方法可用于求解方程，也有助于确定抛物线的顶点。

Consider x² + 6x + 8 = 0. First, rewrite the first two terms as (x + 3)² − 9 + 8 = 0.

考虑x² + 6x + 8 = 0。首先将前两项改写为(x + 3)² − 9 + 8 = 0。

(x + 3)² = 1

Then take square roots: x + 3 = ±1, so x = −2 or x = −4.

然后开平方：x + 3 = ±1，所以x = −2或x = −4。

The same technique can express a quadratic in vertex form y = a(x − h)² + k, where (h, k) is the vertex.

同样的技巧可以把二次函数写成顶点式y = a(x − h)² + k，其中(h, k)是顶点坐标。

5. The Quadratic Formula | 二次公式

The quadratic formula is the most reliable method for solving any quadratic equation. For ax² + bx + c = 0, the solutions are

二次公式是求解任意二次方程最可靠的方法。对于ax² + bx + c = 0，解为

x = [−b ± √(b² − 4ac)] / (2a)

This formula works even when the roots are irrational or complex. It is derived by completing the square on the general quadratic equation.

即使根为无理数或复数，该公式也适用。它是通过对一般二次方程配方推导出来的。

Example: Solve 2x² − 4x − 1 = 0. Here a = 2, b = −4 and c = −1.

例如：解2x² − 4x − 1 = 0。这里a = 2，b = −4，c = −1。

Δ = (−4)² − 4 × 2 × (−1) = 16 + 8 = 24

The roots are x = (4 ± √24) / 4 = (2 ± √6) / 2.

根为x = (4 ± √24) / 4 = (2 ± √6) / 2。

6. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root sign.

判别式是二次公式中根号下的部分。

Δ = b² − 4ac

The value of Δ tells us how many real roots the quadratic equation has.

Δ的值告诉我们二次方程有多少个实数根。

Discriminant / 判别式	Number of Real Roots / 实数根个数	Graph Intersection with x-axis / 图像与x轴交点
Δ > 0	2 distinct real roots 两个不同实根	Intersects at two points 与x轴交于两点
Δ = 0	1 repeated real root 一个重根	Touches at one point 与x轴相切于一点
Δ < 0	0 real roots 没有实数根	Does not intersect the x-axis 不与x轴相交

For example, the equation x² − 2x + 1 = 0 has Δ = 0, so it has exactly one repeated root x = 1.

例如，方程x² − 2x + 1 = 0的Δ = 0，因此它只有一个重根x = 1。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

y = ax² + bx + c的图像是一条抛物线。若a > 0，抛物线开口向上；若a < 0，开口向下。

The axis of symmetry is a vertical line passing through the vertex.

对称轴是经过顶点的一条竖直线。

x = −b / (2a)

The vertex can be found using the axis of symmetry, or by completing the square.

顶点可以通过对称轴公式求得，也可以通过配方确定。

(−b/(2a), c − b²/(4a))

If Δ > 0, the parabola crosses the x-axis twice. If Δ = 0, it touches the x-axis once. If Δ < 0, it stays entirely above or below the x-axis.

若Δ > 0，抛物线与x轴相交两次；若Δ = 0，它与x轴相切一次；若Δ < 0，则抛物线完全位于x轴上方或下方。

8. Relationships Between Roots and Coefficients | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots can be read directly from the coefficients.

对于二次方程ax² + bx + c = 0，若其根为α和β，则根的和与积可以直接从系数中读出。

α + β = −b / a

αβ = c / a

These relationships are useful when we need to find expressions such as α² + β² or 1/α + 1/β without solving the equation fully.

这些关系常用于求解α² + β²或1/α + 1/β等表达式，而无需完整解出方程。

For example, for x² − 3x + 2 = 0, the roots are 1 and 2, so the sum is 3 and the product is 2, matching −b/a = 3 and c/a = 2.

例如，对于x² − 3x + 2 = 0，根为1和2，所以和为3，积为2，与−b/a = 3和c/a = 2一致。

9. Word Problems with Quadratic Equations | 二次方程应用题

Many real-life problems can be solved by forming a quadratic equation. The usual steps are:

许多实际问题可以通过建立二次方程来求解。通常的步骤如下：

Define the unknown variable and express the given conditions as an equation.

定义未知变量，并将题目条件表示为方程。
Simplify the equation into the standard form ax² + bx + c = 0.

将方程化简为标准形式ax² + bx + c = 0。
Solve the equation using factorisation or the quadratic formula.

使用因式分解或二次公式求解方程。
Check whether each solution makes sense within the problem context.

检查每个解是否符合题目实际背景。

For example, a rectangle has length 3 cm more than its width and an area of 54 cm². If the width is x cm, then x(x + 3) = 54.

例如，一个长方形的长比宽多3厘米，面积为54平方厘米。设宽为x厘米，则x(x + 3) = 54。

x² + 3x − 54 = 0

Factorising gives (x + 9)(x − 6) = 0, so x = 6 or x = −9. Since width cannot be negative, the width is 6 cm.

因式分解得(x + 9)(x − 6) = 0，所以x = 6或x = −9。由于宽不可能为负，因此宽为6厘米。

10. Common Mistakes and Examination Tips | 常见错误与考试建议

Students often lose marks on quadratic equations because of small sign errors or careless arithmetic. Here are some common mistakes to avoid:

学生在二次方程题目中经常因为符号错误或粗

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mastering Quadratic Equations | 精通二次方程