Solving Quadratic Equations | 求解二次方程

📚 Solving Quadratic Equations | 求解二次方程

Quadratic equations appear frequently in IGCSE mathematics. Learning how to solve them reliably gives you access to a wide range of exam questions, from simple algebra to geometry and real-life problems.

二次方程在IGCSE数学中频繁出现。学会可靠地解二次方程，能让你应对从简单代数到几何以及实际生活问题中的大量考试题型。

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. The name comes from the Latin word ‘quadratus’, meaning square, because the highest power of x is 2.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a ≠ 0。名称源自拉丁语 ‘quadratus’，意思为平方，因为x的最高次数是2。

If a = 0, the equation becomes linear, not quadratic. In IGCSE exams, you must always check that the coefficient of x² is not zero before identifying a quadratic equation.

如果 a = 0，方程就变成一次方程而不是二次方程。在IGCSE考试中，你必须先检查x²的系数不是零，才能识别为二次方程。

The solutions to a quadratic equation are also called its roots or zeros. They are the x-values that make the equation true.

二次方程的解也称为根或零点。它们是使方程成立的x值。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer roots. You write the quadratic as a product of two linear factors, then set each factor equal to zero.

当二次方程具有简单的整数根时，因式分解是最快的方法。你将二次式写成两个一次因式的乘积，然后令每个因式为零。

Example: Solve x² − 5x + 6 = 0.

示例：解 x² − 5x + 6 = 0。

Find two numbers that multiply to give 6 and add to give −5. The numbers are −2 and −3, so:

找到两个数，相乘得6，相加得−5。这两个数是−2和−3，因此：

(x − 2)(x − 3) = 0

Then x − 2 = 0 gives x = 2, and x − 3 = 0 gives x = 3. Always check your roots by substituting them back into the original equation.

于是 x − 2 = 0 得到 x = 2，x − 3 = 0 得到 x = 3。始终通过代回原方程来检验你的根。

If the coefficient of x² is not 1, you may need to factorise in the form (px + q)(rx + s). For example, 2x² + 7x + 3 = (2x + 1)(x + 3).

如果x²的系数不是1，你可能需要将其分解为 (px + q)(rx + s) 的形式。例如，2x² + 7x + 3 = (2x + 1)(x + 3)。

When an equation has a common factor, take it out first. For example, 3x² − 12x = 0 becomes 3x(x − 4) = 0, so x = 0 or x = 4.

当方程含有公因数时，先提取公因数。例如，3x² − 12x = 0 变为 3x(x − 4) = 0，所以 x = 0 或 x = 4。

3. The Quadratic Formula | 二次公式

Some quadratics cannot be factorised easily. In such cases, the quadratic formula works for any quadratic equation of the form ax² + bx + c = 0.

有些二次式不容易因式分解。在这种情况下，二次公式适用于任何形式为 ax² + bx + c = 0 的二次方程。

x = (−b ± √(b² − 4ac)) / (2a)

The symbol ± means that you take two separate values: one with the plus sign and one with the minus sign.

符号 ± 表示你需要计算两个独立的值：一个取加号，一个取减号。

Example: Solve 2x² + 5x − 3 = 0 using the quadratic formula.

示例：使用二次公式解 2x² + 5x − 3 = 0。

Here a = 2, b = 5 and c = −3. Substitute into the formula:

这里 a = 2，b = 5，c = −3。代入公式：

x = (−5 ± √(5² − 4 × 2 × (−3))) / (2 × 2)

x = (−5 ± √(25 + 24)) / 4 = (−5 ± √49) / 4 = (−5 ± 7) / 4

So x = (−5 + 7) / 4 = 2/4 = 1/2, or x = (−5 − 7) / 4 = −12/4 = −3. Write your answers as a set: x = 1/2 or x = −3.

因此 x = (−5 + 7) / 4 = 2/4 = 1/2，或 x = (−5 − 7) / 4 = −12/4 = −3。写成集合形式：x = 1/2 或 x = −3。

On the IGCSE calculator paper, you can use the calculator’s polynomial solver to check, but you must show the formula method clearly for full marks.

在IGCSE允许使用计算器的试卷上，你可以用计算器的多项式求解功能来检查，但为了得满分，你必须清楚地展示公式法的步骤。

4. Completing the Square | 配方法

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

配方法将二次式改写为 (x + p)² + q 的形式。这在求二次图像的顶点以及解方程时特别有用。

For the equation x² + bx + c = 0, you can write:

对于方程 x² + bx + c = 0，你可以写成：

x² + bx + c = (x + b/2)² − (b/2)² + c

Example: Solve x² + 6x − 7 = 0 by completing the square.

示例：用配方法解 x² + 6x − 7 = 0。

Here b = 6, so b/2 = 3. Thus:

这里 b = 6，所以 b/2 = 3。因此：

(x + 3)² − 9 − 7 = 0

(x + 3)² = 16

Take square roots of both sides: x + 3 = ±4, so x = 1 or x = −7.

两边开平方根：x + 3 = ±4，所以 x = 1 或 x = −7。

If the coefficient of x² is not 1, factor it out first. For example, 2x² + 4x − 3 = 0 becomes 2(x² + 2x) − 3 = 0, then complete the square inside the bracket.

如果x²的系数不是1，先将其提取出来。例如，2x² + 4x − 3 = 0 变为 2(x² + 2x) − 3 = 0，然后在括号内配方。

5. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant, often written as Δ (Greek letter delta).

二次公式中的表达式 b² − 4ac 被称为判别式，通常写作 Δ（希腊字母delta）。

Δ = b² − 4ac

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

Δ的值告诉我们方程有多少个实根：

If Δ > 0: two distinct real roots. For example, x² − 3x + 2 = 0 has Δ = 1, so two roots.

如果 Δ > 0：有两个不同的实根。例如，x² − 3x + 2 = 0 的 Δ = 1，因此有两个根。
If Δ = 0: exactly one real root, also called a repeated root. For example, x² − 6x + 9 = 0 has Δ = 0, so x = 3 only.

如果 Δ = 0：恰好有一个实根，也称为重根。例如，x² − 6x + 9 = 0 的 Δ = 0，所以只有 x = 3。
If Δ < 0: no real roots. The graph does not cross the x-axis. For example, x² + 2x + 5 = 0 has Δ = −16.

如果 Δ < 0：没有实根。图像不穿过x轴。例如，x² + 2x + 5 = 0 的 Δ = −16。

In IGCSE questions, you may be asked to “use the discriminant” to determine the nature of roots or to find a range of values for k in equations like x² + kx + 4 = 0.

在IGCSE题目中，你可能被要求“使用判别式”来确定根的性质，或解像 x² + kx + 4 = 0 这样含参数k的方程中k的取值范围。

6. Quadratic Graphs | 二次函数图像

A quadratic function y = ax² + bx + c produces a parabola. If a > 0, the parabola opens upward (a ‘smile’). If a < 0, it opens downward (a 'frown').

二次函数 y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上（像“微笑”）。如果 a < 0，抛物线开口向下（像“皱眉”）。

The roots of the equation are the x-coordinates where the parabola crosses or touches the x-axis. The discriminant determines the number of intersections.

方程的根是抛物线与x轴相交或相切点的x坐标。判别式决定交点的个数。

The y-intercept is always the constant term c, because when x = 0, y = c.

y轴截距总是常数项 c，因为当 x = 0 时，y = c。

The turning point (vertex) can be found by completing the square. For y = (x − p)² + q, the vertex is (p, q). Alternatively, use x = −b / (2a) for the axis of symmetry.

顶点（极值点）可以通过配方求得。对于 y = (x − p)² + q，顶点为 (p, q)。或者，用 x = −b / (2a) 求对称轴。

For example, y = (x − 1)² − 4 has vertex (1, −4). Setting y = 0 gives (x − 1)² = 4, so x = −1 or x = 3.

例如，y = (x − 1)² − 4 的顶点是 (1, −4)。令 y = 0 得到 (x − 1)² = 4，所以 x = −1 或 x = 3。

7. Word Problems | 应用题

Quadratic equations often arise in area problems, projectile motion and number problems. You must translate the English description into an equation carefully.

二次方程常常出现在面积问题、抛体运动和数字问题中。你必须仔细将文字描述转化为方程。

Example: The area of a rectangle is 40 cm². Its length is 3 cm more than its width. Find the width.

示例：矩形的面积是40 cm²。它的长比宽多3 cm。求宽。

Let the width be x cm. Then the length is (x + 3) cm. So x(x + 3) = 40, which gives x² + 3x − 40 = 0.

设宽为 x cm。那么长为 (x + 3) cm。因此 x(x + 3) = 40，即 x² + 3x − 40 = 0。

Factorise: (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since the width cannot be negative, the width is 5 cm.

因式分解：(x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于宽不能为负，因此宽度为5 cm。

Always reject negative solutions in geometry problems, and state the correct units in your final answer.

在几何问题中始终舍去负数解，并在最终答案中写明正确的单位。

8. Common Mistakes | 常见错误

Many students lose marks on quadratic equations because of avoidable errors. Here are the most frequent pitfalls.

许多学生在二次方程上丢分是因为可以避免的错误。以下是最常见的陷阱。

Not writing the equation in the form ax² + bx + c = 0 before factorising or using the formula.

在因式分解或使用公式之前，没有将方程整理成 ax² + bx + c = 0 的形式。
Forgetting that the quadratic formula has a negative b term. For example, x = (−b ± …) not x = (b ± …).

忘记二次公式中的 −b 项。例如，x = (−b ± …) 而不是 x = (b ± …)。
Sign errors when substituting negative coefficients. Always put brackets around negative values.

代入负系数时出现符号错误。始终用括号将负值括起来。
Dividing both sides by x without considering x = 0. For example, x² = 5x; dividing by x gives x = 5, losing x = 0.

两边同时除以x时没有考虑 x = 0。例如，x² = 5x；除以x得到 x = 5，丢掉了 x = 0。
Not verifying roots. Substitute your answers back into the original equation to catch calculation slips.

不检验根。把答案代回原方程，以发现计算错误。

Practice these methods on different examples until the steps become automatic.

用不同例题反复练习这些方法，直到步骤变得熟练自然为止。

9. Summary | 总结

There are three main methods to solve quadratic equations: factorisation, the quadratic formula, and completing the square. Each has its advantages.

解二次方程有三种主要方法：因式分解、二次公式和配方法。每种方法各有优势。

Factorisation is quick when the roots are rational. The quadratic formula always works. Completing the square helps with graphs and harder algebraic manipulation.

当根为有理数时，因式分解很快。二次公式始终有效。配方法有助于处理图像和更复杂的代数变形。

The discriminant b² − 4ac tells you the number of real roots without solving the equation completely.

判别式 b² − 4ac 可以告诉你实根的个数，而不必完整求解方程。

Always check your answers, and remember that in many word problems negative roots must be rejected based on the context.

始终检查你的答案，并记住在许多应用题中，负数根必须根据实际情况舍去。

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Solving Quadratic Equations | 求解二次方程