Quadratic Equations | 二次方程

📚 Quadratic Equations | 二次方程

Quadratic equations are polynomial equations of degree two. They appear throughout IGCSE Mathematics and in many real-world problems.

二次方程是二次多项式方程，在IGCSE数学以及许多实际问题中经常出现。

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1. Standard Form | 标准形式

A quadratic equation in one variable can be written as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

一元二次方程可以写成 ax² + bx + c = 0 的形式，其中 a、b、c 为常数，且 a ≠ 0。

Here are some examples:

以下是一些例子：

• x² − 5x + 6 = 0

  x² − 5x + 6 = 0

• 2x² + 3x − 1 = 0

  2x² + 3x − 1 = 0

• −x² + 4x = 0

  −x² + 4x = 0

Notice that the highest power of x is 2, which makes it quadratic.

注意 x 的最高次数为 2，因此它是二次的。

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2. Factorisation Method | 因式分解法

If the quadratic expression can be factorised, we use the zero-product property: if pq = 0, then p = 0 or q = 0.

如果二次多项式可以因式分解，我们利用零乘积性质：若 pq = 0，则 p = 0 或 q = 0。

Follow these steps:

按照以下步骤：

• Write the equation in standard form ax² + bx + c = 0.

  把方程写成标准形式 ax² + bx + c = 0。

• Factorise the left-hand side into two linear factors.

  把左边分解为两个一次因式。

• Set each factor equal to zero and solve for x.

  令每个因式等于零，然后解出 x。

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3. Worked Example | 例题详解

Solve the equation x² − 5x + 6 = 0.

解方程 x² − 5x + 6 = 0。

x² − 5x + 6 = 0

Factorise: (x − 2)(x − 3) = 0.

因式分解：(x − 2)(x − 3) = 0。

Using the zero-product property: x − 2 = 0 or x − 3 = 0.

利用零乘积性质：x − 2 = 0 或 x − 3 = 0。

So the solutions are x = 2 or x = 3.

因此解为 x = 2 或 x = 3。

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4. The Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by:

对于任意二次方程 ax² + bx + c = 0，解由下式给出：

x = (−b ± √b² − 4ac) / (2a)

This formula works for all quadratics, including those that cannot be factorised easily.

这个公式适用于所有二次方程，包括不容易因式分解的方程。

Be careful with signs and the order of operations.

注意符号和运算顺序。

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5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q, which is useful for solving and graphing.

配方法将二次式改写为 a(x + p)² + q 的形式，这对求解和作图很有用。

For a monic quadratic x² + bx, add (b/2)² to make a perfect square.

对于首项系数为 1 的二次式 x² + bx，加上 (b/2)² 即可配成完全平方。

Example: x² + 6x = (x + 3)² − 9.

例如：x² + 6x = (x + 3)² − 9。

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6. The Discriminant | 判别式

The discriminant is Δ = b² − 4ac. It tells us the nature of the roots without solving.

判别式为 Δ = b² − 4ac。它可以帮助我们不解方程就能判断根的性质。

Δ = b² − 4ac	Nature of roots | 根的性质
Δ > 0	Two distinct real roots | 两个不相等的实数根
Δ = 0	One repeated root | 两个相等的实数根（重根）
Δ < 0	No real roots | 没有实数根

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7. Solving by Completing the Square | 用配方法解方程

Solve x² + 6x + 2 = 0 by completing the square.

用配方法解 x² + 6x + 2 = 0。

First move the constant: x² + 6x = −2.

首先移常数项：x² + 6x = −2。

Add (6/2)² = 9 to both sides: x² + 6x + 9 = 7.

两边加上 (6/2)² = 9：x² + 6x + 9 = 7。

This gives (x + 3)² = 7.

于是得到 (x + 3)² = 7。

Take square roots: x + 3 = ±√7.

开平方：x + 3 = ±√7。

Hence x = −3 ± √7.

因此 x = −3 ± √7。

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8. Word Problems | 应用题

Quadratic equations often arise from geometry or area problems.

二次方程经常在几何或面积问题中出现。

Example: A rectangle's length is 3 cm longer than its width. Its area is 40 cm². Find the dimensions.

例：一个长方形的长比宽长 3 厘米，面积为 40 平方厘米。求长和宽。

Let width = x, then length = x + 3. Since area = 40, we have x(x + 3) = 40.

设宽为 x，则长为 x + 3。由于面积为 40，所以 x(x + 3) = 40。

x² + 3x − 40 = 0, which factorises to (x + 8)(x − 5) = 0.

x² + 3x − 40 = 0，因式分解为 (x + 8)(x − 5) = 0。

Since x > 0, x = 5. The rectangle is 5 cm by 8 cm.

因为 x > 0，所以 x = 5。长方形为 5 厘米 × 8 厘米。

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9. Graphical Solutions | 图象解法

tions of ax² + bx + c = 0 correspond to the x-intercepts of the parabola y = ax² + bx + c.

方程 ax² + bx + c = 0 的解对应抛物线 y = ax² + bx + c 与 x 轴的交点。

If the parabola crosses the x-axis at two points, the roots are the x-coordinates of those points.

如果抛物线与 x 轴有两个交点，那么根的值为交点的 x 坐标。

If it touches the axis once, there is one repeated root. If it never crosses, there are no real roots.

如果它与 x 轴相切一次，则有一个重根；如果它从未穿过 x 轴，则没有实数根。

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10. Common Mistakes and Tips | 常见错误与提示

Avoid these common pitfalls:

避免以下常见错误：

• Forgetting to rearrange the equation into standard form before factorising.

  因式分解前忘记将方程整理成标准形式。

• Sign errors when using the quadratic formula, especially with negative b or c.

  使用求根公式时出现符号错误，尤其是 b 或 c 为负数时。

• Incorrectly expanding (x + p)²; remember the middle term 2px.

  错误展开 (x + p)²；记住中间项 2px。

• Forgetting the ± symbol when taking square roots.

  开平方时忘记 ± 号。

• In word problems, discarding negative or zero solutions if they do not make sense.

  在应用题中，如果负数或零解不符合实际，则舍去。

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

始终将解代回原方程进行检验。

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