Solving Quadratic Equations: Methods and Graphs | 解二次方程：方法与图像

📚 Solving Quadratic Equations: Methods and Graphs | 解二次方程：方法与图像

Quadratic equations appear frequently in IGCSE Mathematics. Mastering their solution methods and graphical interpretation is essential for exam success.

在 IGCSE 数学中，二次方程非常常见。掌握其解法和图像理解是考试成功的关键。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。

The highest power of the variable x is 2. This is why the graph of a quadratic function is a parabola.

变量 x 的最高次数是 2，因此二次函数的图像是一条抛物线。

For example, 2x² − 5x + 3 = 0 is a quadratic equation, while x³ − 2x = 0 is not.

例如，2x² − 5x + 3 = 0 是二次方程，而 x³ − 2x = 0 不是。

2. Solving by Factorisation | 因式分解法

Factorisation is the simplest method when the quadratic expression can be written as a product of two linear factors.

当二次表达式能写成两个一次因式的乘积时，因式分解是最简单的方法。

For example, solve x² − 5x + 6 = 0. We look for two numbers whose product is 6 and whose sum is −5.

例如，解 x² − 5x + 6 = 0。我们需要找两个数，使得它们的乘积为 6，且和为 −5。

(x − 2)(x − 3) = 0

Using the zero product property, x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

根据零积性质，x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

Always check whether the equation is in standard form before factorising.

因式分解前，一定要先确认方程已经化为标准形式。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This is useful for finding turning points and solving equations.

配方法将二次表达式改写为 a(x + p)² + q 的形式。这有助于求顶点以及解方程。

Consider x² + 6x + 1 = 0. Half of 6 is 3, so we write (x + 3)² − 9 + 1 = 0.

考虑 x² + 6x + 1 = 0。6 的一半是 3，因此写成 (x + 3)² − 9 + 1 = 0。

(x + 3)² − 8 = 0

Then (x + 3)² = 8, so x + 3 = ±√8, and x = −3 ± 2√2.

于是 (x + 3)² = 8，所以 x + 3 = ±√8，即 x = −3 ± 2√2。

If the coefficient of x² is not 1, first factor out that coefficient from the x² and x terms.

如果 x² 的系数不是 1，应先把该系数从 x² 项和 x 项中提出来。

4. The Quadratic Formula | 求根公式

The quadratic formula can solve any quadratic equation, including those that do not factorise easily.

求根公式可以解任何二次方程，包括那些不容易因式分解的方程。

x = (−b ± √(b² − 4ac)) / (2a)

For the equation ax² + bx + c = 0, substitute the values of a, b and c into the formula.

对于方程 ax² + bx + c = 0，将 a、b、c 的值代入公式即可。

Example: Solve 2x² + 3x − 2 = 0 using the formula.

例：用公式法解 2x² + 3x − 2 = 0。

Here a = 2, b = 3, c = −2. The discriminant is 3² − 4 × 2 × (−2) = 9 + 16 = 25.

这里 a = 2，b = 3，c = −2。判别式为 3² − 4 × 2 × (−2) = 9 + 16 = 25。

x = (−3 ± √25) / 4 = (−3 ± 5) / 4

Therefore x = 1/2 or x = −2.

因此 x = 1/2 或 x = −2。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: b² − 4ac.

判别式是求根公式中根号内的部分：b² − 4ac。

If b² − 4ac > 0, the equation has two distinct real roots.

如果 b² − 4ac > 0，方程有两个不相等的实数根。

If b² − 4ac = 0, the equation has exactly one real root (a repeated root).

如果 b² − 4ac = 0，方程有两个相等实数根（一个重根）。

If b² − 4ac < 0, the equation has no real roots.

如果 b² − 4ac < 0，方程没有实数根。

The discriminant also tells us how many x-intercepts the graph has.

判别式也告诉我们图像与 x 轴的交点个数。

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

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

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

The solutions of ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.

ax² + bx + c = 0 的解就是图像与 x 轴交点的横坐标。

The turning point of the parabola can be found by completing the square or using x = −b/(2a).

抛物线的顶点可以通过配方法或 x = −b/(2a) 来求得。

Example: y = x² − 4x + 3 can be written as y = (x − 2)² − 1, so the minimum point is (2, −1).

例如：y = x² − 4x + 3 可写成 y = (x − 2)² − 1，因此最小值点为 (2, −1)。

7. Roots, Factorisation and the Graph | 根、因式分解与图像的关系

If a quadratic equation has roots r and s, then the quadratic can be written as a(x − r)(x − s) = 0.

如果二次方程的两个根为 r 和 s，则该二次方程可写成 a(x − r)(x − s) = 0 的形式。

For example, if the roots are −1 and 5, the equation must contain factors (x + 1) and (x − 5).

例如，若根为 −1 和 5，则方程必须包含因式 (x + 1) 和 (x − 5)。

The axis of symmetry of the parabola is exactly halfway between the roots.

抛物线的对称轴正好位于两个根的正中间。

If the roots are −1 and 5, the axis of symmetry is x = (−1 + 5)/2 = 2.

若根为 −1 和 5，则对称轴为 x = (−1 + 5)/2 = 2。

8. Solving Word Problems with Quadratics | 用二次方程解应用题

Many real-world problems involve quadratic equations, such as area, projectile motion and number problems.

许多实际问题涉及二次方程，例如面积问题、抛体运动问题和数字问题。

Read the problem carefully and define a variable for the unknown quantity.

仔细读题，并为未知量设定一个变量。

Example: The area of a rectangle is 30 cm². Its length is 1 cm greater than its width. Find the width.

例：一个长方形的面积是 30 cm²。它的长比宽多 1 cm。求宽。

Let width = w, then length = w + 1. So w(w + 1) = 30.

设宽为 w，则长为 w + 1。所以 w(w + 1) = 30。

w² + w − 30 = 0

Factorising gives (w + 6)(w − 5) = 0, so w = 5 (reject w = −6 because length cannot be negative).

因式分解得 (w + 6)(w − 5) = 0，所以 w = 5（舍去 w = −6，因为长度不能为负）。

9. Common Mistakes | 常见错误

One common mistake is forgetting to set the equation to zero before factorising.

一个常见错误是忘记在因式分解前将方程化为零的形式。

Another mistake is misapplying the square root property: x² = 9 gives x = ±3, not just x = 3.

另一个错误是错误应用平方根性质：x² = 9 应得到 x = ±3，而不仅仅是 x = 3。

Students also often confuse the sign of b in the quadratic formula, especially when b is negative.

学生在代入求根公式时也常常搞错 b 的符号，尤其是当 b 为负数时。

Always check your solutions by substituting them back into the original equation.

始终将解代回原方程进行检验。

10. Summary and Revision Tips | 总结与复习建议

To solve any quadratic equation, choose the most efficient method:

解任何二次方程时，选择最有效的方法：

Try factorisation first if the roots are integers or simple fractions.

如果根是整数或简单分数，首先尝试因式分解。
Use completing the square when asked to find the turning point.

当要求找顶点时，使用配方法。
Use the quadratic formula when the equation does not factorise easily.

当方程不易因式分解时，使用求根公式。
Use the discriminant to check the number of real roots.

使用判别式判断实数根的个数。

Practise sketching parabolas to understand the connection between equations and graphs.

多加练习抛物线草图，理解方程与图像之间的联系。

With regular practice, solving quadratic equations becomes quick and accurate.

通过定期练习，解二次方程会变得快速而准确。

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Solving Quadratic Equations: Methods and Graphs | 解二次方程：方法与图像