Quadratic Equations | 二次方程

📚 Quadratic Equations | 二次方程

Quadratic equations are among the most important algebraic structures in A-Level mathematics. They appear in pure mathematics, mechanics, statistics, and a wide range of real-world problems. Mastering the methods of solving and analysing quadratics is essential for success in AQA A-Level Mathematics.

二次方程是A-Level数学中最重要的代数结构之一。它们出现在纯数学、力学、统计学以及各类实际问题中。掌握二次方程的求解与分析方法是 AQA A-Level 数学取得高分的关键。

1. Forms of Quadratic Expressions | 二次表达式的形式

A quadratic expression in the variable \(x\) has the standard form \(ax^{2} + bx + c\), where \(a\), \(b\) and \(c\) are real constants and \(a \neq 0\). The name “quadratic” comes from the Latin word “quadratus”, meaning square, because the highest power of the variable is \(x^{2}\).

二次表达式在变量 \(x\) 下的标准形式为 \(ax^{2} + bx + c\)，其中 \(a\)、\(b\)、\(c\) 为实数常数，且 \(a \neq 0\)。”二次”这一名称来自拉丁语 “quadratus”，意为”平方”，因为变量的最高次数是 \(x^{2}\)。

There are three common forms of a quadratic expression.

二次表达式有三种常见形式。

The general form: \(ax^{2} + bx + c\).

一般式：\(ax^{2} + bx + c\)。
The completed square form: \(a(x + p)^{2} + q\), which shows the turning point of the graph.

配方式：\(a(x + p)^{2} + q\)，它直接给出图像的顶点坐标。
The factorised form: \(a(x – \alpha)(x – \beta)\), which shows the roots of the equation.

因式分解式：\(a(x – \alpha)(x – \beta)\)，它直接给出方程的根。

Each form is useful in different situations. For example, the factorised form is best for solving equations, while the completed square form is best for sketching graphs and identifying maximum or minimum values.

每种形式在不同情境中各有用途。例如，因式分解式最适合解方程，而配方式最适合画图以及确定最大值或最小值。

2. Solving by Factorisation | 因式分解法

The simplest method for solving a quadratic equation is factorisation. First, rearrange the equation so that one side is zero. Then factorise the quadratic expression into two linear factors. Finally, use the zero product property: if the product of two factors is zero, then at least one of the factors must be zero.

解二次方程最简单的方法是因式分解。首先，将方程整理为一边等于零的形式；然后，将二次表达式分解为两个一次因式；最后，利用零积性质：若两个因式相乘为零，则至少其中一个因式为零。

Consider the equation \(x^{2} – 5x + 6 = 0\).

考察方程 \(x^{2} – 5x + 6 = 0\)。

\(x^{2} – 5x + 6 = (x – 2)(x – 3) = 0\)

Therefore, \(x – 2 = 0\) or \(x – 3 = 0\), so the solutions are \(x = 2\) and \(x = 3\).

因此，\(x – 2 = 0\) 或 \(x – 3 = 0\)，所以解为 \(x = 2\) 和 \(x = 3\)。

When the coefficient of \(x^{2}\) is not 1, factorisation requires more care. For example, \(2x^{2} + 5x + 2 = 0\) can be written as \((2x + 1)(x + 2) = 0\). The roots are \(x = -\frac{1}{2}\) and \(x = -2\).

当 \(x^{2}\) 的系数不为 1 时，因式分解需要更加细心。例如，\(2x^{2} + 5x + 2 = 0\) 可写成 \((2x + 1)(x + 2) = 0\)。根为 \(x = -\frac{1}{2}\) 和 \(x = -2\)。

Factorisation works only when the roots are rational numbers. If the discriminant is not a perfect square, another method is required.

因式分解仅在根为有理数时适用。如果判别式不是完全平方数，则需要使用其他方法。

3. Solving by the Quadratic Formula | 二次方程求根公式

For any quadratic equation \(ax^{2} + bx + c = 0\), the solutions are given by the quadratic formula. This formula works in all cases, including equations with irrational or complex roots.

对于任意二次方程 \(ax^{2} + bx + c = 0\)，解可由二次求根公式给出。该公式适用于所有情况，包括含无理根或复数根的方程。

\(x = \frac{-b \pm \sqrt{b^{2} – 4ac}}{2a}\)

The symbol \(\pm\) means that there are two possible values: one using the positive square root and one using the negative square root.

符号 \(\pm\) 表示存在两个可能的值：一个取正平方根，另一个取负平方根。

Example: Solve \(2x^{2} + 3x – 5 = 0\). Here \(a = 2\), \(b = 3\), \(c = -5\).

例：解 \(2x^{2} + 3x – 5 = 0\)。这里 \(a = 2\)，\(b = 3\)，\(c = -5\)。

\(x = \frac{-3 \pm \sqrt{3^{2} – 4 \times 2 \times (-5)}}{2 \times 2} = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4}\)

So \(x = 1\) or \(x = -\frac{5}{2}\).

因此 \(x = 1\) 或 \(x = -\frac{5}{2}\)。

Always check whether the square root can be simplified. For instance, \(\sqrt{12} = 2\sqrt{3}\). In AQA A-Level examinations, an unsimplified surd may lose marks.

始终检查平方根是否可以化简。例如，\(\sqrt{12} = 2\sqrt{3}\)。在 AQA A-Level 考试中，未化简的根式可能会被扣分。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic expression as a perfect square plus a constant. This method is particularly useful for finding turning points, solving equations, and deriving the quadratic formula itself.

配方法将二次表达式改写为一个完全平方加常数。这一方法在求顶点坐标、解方程以及推导求根公式时尤为有用。

Consider \(x^{2} + 6x + 1 = 0\). First, take half of the coefficient of \(x\), which is 3, and square it to get 9.

考察 \(x^{2} + 6x + 1 = 0\)。首先，取 \(x\) 系数的一半，即 3，平方后得到 9。

\((x + 3)^{2} – 9 + 1 = 0\)

This simplifies to \((x + 3)^{2} = 8\). Therefore, \(x + 3 = \pm \sqrt{8}\), so \(x = -3 \pm 2\sqrt{2}\).

化简得 \((x + 3)^{2} = 8\)。因此，\(x + 3 = \pm \sqrt{8}\)，所以 \(x = -3 \pm 2\sqrt{2}\)。

For the general expression \(ax^{2} + bx + c\), completing the square gives:

对于一般表达式 \(ax^{2} + bx + c\)，配方可得：

\(a\left(x + \frac{b}{2a}\right)^{2} + c – \frac{b^{2}}{4a}\)

The turning point of the graph \(y = ax^{2} + bx + c\) is therefore at \(\left(-\frac{b}{2a}, c – \frac{b^{2}}{4a}\right)\).

因此，图像 \(y = ax^{2} + bx + c\) 的顶点坐标为 \(\left(-\frac{b}{2a}, c – \frac{b^{2}}{4a}\right)\)。

5. The Discriminant | 判别式

The expression \(b^{2} – 4ac\) inside the square root of the quadratic formula is called the discriminant, denoted by \(\Delta\). It determines the nature of the roots of a quadratic equation.

求根公式根号内的表达式 \(b^{2} – 4ac\) 称为判别式，记作 \(\Delta\)。它决定了二次方程根的性质。

Value of \(\Delta\)	Nature of roots	Graph interpretation
\(\Delta > 0\)	Two distinct real roots	Graph crosses the x-axis twice
\(\Delta = 0\)	One repeated real root	Graph touches the x-axis once
\(\Delta < 0\)	No real roots	Graph does not meet the x-axis

For example, the equation \(x^{2} + 4x + 5 = 0\) has discriminant \(\Delta = 4^{2} – 4 \times 1 \times 5 = -4\). Since \(\Delta < 0\), the equation has no real solutions.

例如，方程 \(x^{2} + 4x + 5 = 0\) 的判别式为 \(\Delta = 4^{2} – 4 \times 1 \times 5 = -4\)。由于 \(\Delta < 0\)，该方程没有实数解。

In AQA A-Level Mathematics, you may also need to use the discriminant to determine how many intersections exist between a line and a curve. Substituting the equation of the line into the curve always produces a quadratic equation whose discriminant tells you the number of intersection points.

在 AQA A-Level 数学中，你可能还需要使用判别式来确定直线与曲线有多少个交点。将直线方程代入曲线方程后总会得到一个二次方程，其判别式可反映交点的个数。

6. Roots and Coefficients | 根与系数的关系

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(ax^{2} + bx + c = 0\), then the sum and product of the roots can be written directly in terms of the coefficients.

若 \(\alpha\) 和 \(\beta\) 是二次方程 \(ax^{2} + bx + c = 0\) 的两个根，则根的和与积可直接用系数表示。

\(\alpha + \beta = -\frac{b}{a}\), \(\quad \alpha\beta = \frac{c}{a}\)

These relationships are useful for constructing quadratic equations from known roots. If a quadratic equation has roots \(\alpha\) and \(\beta\), the equation can be written as:

这些关系可用于由已知根构造二次方程。若二次方程的根为 \(\alpha\) 和 \(\beta\)，方程可写成：

\(x^{2} – (\alpha + \beta)x + \alpha\beta = 0\)

Example: A quadratic equation has roots \(3\) and \(-7\). The sum of the roots is \(-4\), and the product is \(-21\). Therefore, the equation is \(x^{2} + 4x – 21 = 0\).

例：某二次方程的两个根为 \(3\) 和 \(-7\)。根的和为 \(-4\)，根的积为 \(-21\)。因此，方程为 \(x^{2} + 4x – 21 = 0\)。

These relationships also allow you to evaluate expressions such as \(\alpha^{2} + \beta^{2}\) without solving the equation. Since \(\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} – 2\alpha\beta\), you can substitute the coefficient relationships directly.

这些关系还允许你在不解方程的情况下计算 \(\alpha^{2} + \beta^{2}\) 等表达式的值。因为 \(\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} – 2\alpha\beta\)，你可以直接代入系数关系。

7. Quadratic Graphs and Transformations | 二次函数图像与变换

The graph of \(y = ax^{2} + bx + c\) is a parabola. If \(a > 0\), the parabola opens upwards and has a minimum point. If \(a < 0\), the parabola opens downwards and has a maximum point.

\(y = ax^{2} + bx + c\) 的图像是一条抛物线。若 \(a > 0\)，抛物线开口向上，并有最小值点；若 \(a < 0\)，抛物线开口向下，并有最大值点。

The vertex of the parabola is at the point \(\left(-\frac{b}{2a}, c – \frac{b^{2}}{4a}\right)\), which can also be read directly from the completed square form \(y = a(x + p)^{2} + q\). The vertex is then \((-p, q)\).

抛物线的顶点坐标为 \(\left(-\frac{b}{2a}, c – \frac{b^{2}}{4a}\right)\)，也可以直接从配方式 \(y = a(x + p)^{2} + q\) 中读出，即顶点为 \((-p, q)\)。

The y-intercept is \(c\), because substituting \(x = 0\) gives \(y = c\). The x-intercepts, if they exist, are the roots of \(ax^{2} + bx + c = 0\).

由于代入 \(x = 0\) 得 \(y = c\)，所以 y 轴截距为 \(c\)。x 轴截距（若存在）就是方程 \(ax^{2} + bx + c = 0\) 的根。

Simple transformations also apply. For a function \(y = f(x)\), the graph of \(y = f(x) + k\) is shifted upwards by \(k\) units, while \(y = f(x + k)\) is shifted left by \(k\) units. Reflection in the x-axis is represented by \(y = -f(x)\).

简单变换同样适用。对于函数 \(y = f(x)\)，图像 \(y = f(x) + k\) 向上平移 \(k\) 个单位，而 \(y = f(x + k)\) 向左平移 \(k\) 个单位。关于 x 轴的翻折表示为 \(y = -f(x)\)。

When sketching a quadratic graph, always label the vertex, the y-intercept, and any real x-intercepts. If there are no real roots, state this clearly instead of drawing a graph that crosses the x-axis.

画二次函数图像时，务必标出顶点、y 轴截距以及任何实数 x 轴截距。若没有实数根，应明确说明，而不是画出穿过 x 轴的图像。

8. Quadratic Inequalities | 二次不等式

Solving a quadratic inequality such as \(ax^{2} + bx + c > 0\) requires finding the critical values where the expression equals zero, then using a sketch or sign table to determine the intervals where the inequality is satisfied.

解二次不等式（如 \(ax^{2} + bx + c > 0\)）需要先求出表达式等于零时的临界值，再利用草图或符号表确定满足不等式的区间。

Example: Solve \(x^{2} – x – 6 > 0\). First factorise:

例：解 \(x^{2} – x – 6 > 0\)。先因式分解：

\((x – 3)(x + 2) > 0\)

The critical values are \(x = 3\) and \(x = -2\). The coefficient of \(x^{2}\) is positive, so the parabola opens upwards. The expression is positive outside the interval between the roots, therefore:

临界值为 \(x = 3\) 和 \(x = -2\)。\(x^{2}\) 的系数为正，因此抛物线开口向上。表达式在两根区间之外为正，因此：

\(x < -2\) or \(x > 3\)

For a non-strict inequality such as \(x^{2} – x – 6 \geq 0\), the endpoints are included, so the solution is \(x \leq -2\) or \(x \geq 3\).

对于非严格不等式，如 \(x^{2} – x – 6 \geq 0\)，端点包含在内，因此解为 \(x \leq -2\) 或 \(x \geq 3\)。

If a quadratic has discriminant \(\Delta < 0\) and \(a > 0\), then the expression is always positive for all real \(x\). Conversely, if \(a < 0\), the expression is always negative.

若二次式的判别式 \(\Delta < 0\) 且 \(a > 0\)，则表达式对所有实数 \(x\) 恒为正。反之，若 \(a < 0\)，则表达式恒为负。

9. Modelling with Quadratics | 二次函数建模

Quadratics are widely used to model real-world situations involving areas, revenue, projectile motion, and optimisation problems. In AQA A-Level exams, you should be comfortable translating a worded problem into a quadratic equation.

二次函数广泛用于建模涉及面积、收入、抛体运动和优化问题的实际情境。在 AQA A-Level 考试中，你应该能够将文字问题转化为二次方程。

Example: A rectangle has a perimeter of 20 metres. If one side has length \(x\), the other side has length \(10 – x\), so the area is:

例：一个矩形的周长为 20 米。若一边长为 \(x\)，则另一边长为 \(10 – x\)，因此面积为：

\(A = x(10 – x) = 10x – x^{2}\)

This is a quadratic function opening downwards. Its maximum occurs at \(x = 5\), giving a maximum area of 25 square metres, which is the square case.

这是一个开口向下的二次函数。其最大值出现在 \(x = 5\)，最大面积为 25 平方米，此时为正方形。

In mechanics, the height of a projectile is often modelled by \(h = -\frac{1}{2}gt^{2} + vt + h_{0}\), where \(g\) is the acceleration due to gravity, \(v\) is the initial velocity, and \(h_{0}\) is the initial height. Solving \(h = 0\) gives the time when the projectile hits the ground.

在力学中，抛射体的高度常建模为 \(h = -\frac{1}{2}gt^{2} + vt + h_{0}\)，其中 \(g\) 为重力加速度，\(v\) 为初速度，\(h_{0}\) 为初始高度。解 \(h = 0\) 可得到抛射体落地的时间。

When modelling, always state the domain of the variable. In many problems, lengths and times cannot be negative, so you must discard any negative or unreasonable roots.

建模时，务必说明变量的定义域。在许多问题中，长度和时间不能为负，因此必须舍去负数或不合理的根。

10. Exam Tips and Common Mistakes | 考试要点与常见错误

Quadratic equations appear in nearly every AQA A-Level Mathematics paper. A small error in sign or simplification can cost several marks, so accuracy and checking are essential.

二次方程几乎出现在每一份 AQA A-Level 数学试卷中。符号或化简上的一个小错误可能损失数分，因此准确性和检查至关重要。

Do not forget the \(\pm\) symbol when using the quadratic formula. Many students lose marks by writing only one solution.

使用求根公式时不要遗漏 \(\pm\) 符号。许多学生只写一个解而失分。
Do not divide both sides of an equation by \(x\) unless you are sure \(x \neq 0\). Doing so can lose the root \(x = 0\).

除非确定 \(x \neq 0\)，否则不要将方程两边同时除以 \(x\)。这样做可能丢失根 \(x = 0\)。
Always rearrange the equation into the form \(ax^{2} + bx + c = 0\) before applying the formula or factorisation.

在套用公式或因式分解前，务必先将方程整理为 \(ax^{2} + bx + c = 0\) 的形式。
When completing the square, take half of the coefficient of \(x\) and square it. A common error is forgetting to subtract the squared value afterwards.

配方时，要取 \(x\) 系数的一半再平方。常见错误是之后忘记减去该平方值。
When solving inequalities, remember that multiplying or dividing by a negative number reverses the inequality direction. For quadratic inequalities, use a sketch to check the solution region.

解不等式时，要记住乘以或除以负数会改变不等号方向。对于二次不等式，应画草图检查解区间。
Always check your answers by substituting them back into the original equation. This takes only a few seconds and can prevent careless mistakes.

始终将答案代回原方程进行验证。这只需几秒钟，却可以避免粗心错误。

In questions that require answers in exact form, use surds or fractions rather than rounded decimals. In questions that ask for approximate values, round to an appropriate degree of accuracy such as 3 significant figures.

在要求精确答案的问题中，应使用根式或分数，而不是四舍五入的小数。在要求近似值的问题中，应按适当的精度取整，例如保留 3 位有效数字。

Finally, memorise the quadratic formula and the discriminant conditions. Quick recall allows you to focus on the structure of the question rather than on basic algebra.

最后，牢记求根公式和判别式的条件。快速回忆这些内容能让你专注于题目结构，而不是基础代数运算。

Quadratic equations are a central topic in AQA A-Level Mathematics. A clear understanding of factorisation, the quadratic formula, completing the square, the discriminant, and graph sketching will support almost every other area of the course, including trigonometry, calculus, sequences, and mechanics.

二次方程是 AQA A-Level 数学的核心主题。深入理解因式分解、求根公式、配方法、判别式和图像画法，将帮助你掌握课程中几乎所有其他领域

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Quadratic Equations | 二次方程