Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in Paper 2, Paper 4, and in practical problem-solving questions. A solid understanding of how to solve them is essential for achieving a high grade.

二次方程是IGCSE数学中最重要的主题之一。它们出现在试卷2、试卷4以及实际应用题中。扎实掌握解法对于取得高分至关重要。

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

A quadratic equation is an equation where the highest power of the unknown variable is 2. It contains an x² term, and it may also contain an x term and a constant term.

二次方程是未知数的最高次数为2的方程。它包含x²项，也可能包含x项和常数项。

• Examples: x² − 5x + 6 = 0 and 2x² + 3x − 1 = 0 are quadratic equations.
• 例子：x² − 5x + 6 = 0 和 2x² + 3x − 1 = 0 都是二次方程。
• x³ + 2x = 0 is a cubic equation, not quadratic.
• x³ + 2x = 0 是三次方程，不是二次方程。

Notice that the coefficient of x² cannot be zero; if it were zero, the equation would become linear.

注意x²的系数不能为零；如果为零，方程就变成了线性方程。

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

The standard form of a quadratic equation is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程的标准形式写为 ax² + bx + c = 0，其中a、b、c是常数，且a ≠ 0。

ax² + bx + c = 0 (a ≠ 0)

Before solving, you should always rearrange the equation into this form. For example, x² = 4x + 5 must be rewritten as x² − 4x − 5 = 0.

在解方程之前，应始终将方程整理成这种形式。例如，x² = 4x + 5 必须改写为 x² − 4x − 5 = 0。

Equation | 方程	a	b	c
x² − 5x + 6 = 0	1	−5	6
2x² + 3x − 1 = 0	2	3	−1
x² = 4x + 5 → x² − 4x − 5 = 0	1	−4	−5

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3. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic expression can be written as a product of two linear brackets. The rule is: if the product of two expressions is zero, then at least one of them must be zero.

当二次表达式可以写成两个一次括号的乘积时，因式分解是最快的方法。规则是：如果两个表达式的乘积为零，那么其中至少有一个必须为零。

Example | 例子: Solve x² − 5x + 6 = 0.

• Find two numbers that multiply to give 6 and add to give −5: they are −2 and −3.
• 找到两个数，乘积为6，和为−5：它们是−2和−3。
• Write the brackets: (x − 2)(x − 3) = 0.
• 写出括号：(x − 2)(x − 3) = 0。
• Set each bracket to zero: x − 2 = 0 or x − 3 = 0.
• 令每个括号为零：x − 2 = 0 或 x − 3 = 0。
• Solutions: x = 2 or x = 3.
• 解为：x = 2 或 x = 3。

Another common pattern is the difference of two squares: x² − 9 = (x − 3)(x + 3). This gives solutions x = 3 and x = −3.

另一个常见模式是平方差：x² − 9 = (x − 3)(x + 3)。这给出解 x = 3 和 x = −3。

x² − 9 = 0 → (x − 3)(x + 3) = 0 → x = 3 or x = −3

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4. Solving by the Quadratic Formula | 求根公式法

When factorisation is difficult or impossible, use the quadratic formula. This formula works for any quadratic equation in standard form.

当因式分解困难或不适用时，使用求根公式。该公式适用于任何标准形式的二次方程。

x = (−b ± √(b² − 4ac)) / (2a)

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

• Here a = 2, b = 3, c = −2.
• 这里 a = 2，b = 3，c = −2。
• Substitute into the formula: x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2).
• 代入公式：x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2)。
• Simplify: x = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4.
• 化简：x = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4。
• So x = (2) / 4 = 0.5 or x = (−8) / 4 = −2.
• 所以 x = 2/4 = 0.5 或 x = −8/4 = −2。

Always write down the substitution clearly; many exam marks are given for method, not just the final answer.

务必清楚地写出代入过程；考试中很多分数是给方法的，而不只是最终答案。

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5. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method is especially useful for finding the vertex of a parabola or solving equations without using the formula.

配方法将二次式改写为 (x + p)² + q 的形式。这种方法在求抛物线顶点或不用公式解方程时特别有用。

x² + bx + c = (x + b/2)² − (b/2)² + c

Example | 例子: Solve x² + 6x + 4 = 0 by completing the square.

• Take half of 6, which is 3, and write: (x + 3)² − 9 + 4 = 0.
• 取6的一半，即3，写出：(x + 3)² − 9 + 4 = 0。
• Simplify: (x + 3)² − 5 = 0.
• 化简：(x + 3)² − 5 = 0。
• Rearrange: (x + 3)² = 5.
• 移项：(x + 3)² = 5。
• Take the square root: x + 3 = ±√5.
• 取平方根：x + 3 = ±√5。
• Solutions: x = −3 + √5 or x = −3 − √5.
• 解为：x = −3 + √5 或 x = −3 − √5。

Remember that taking a square root always gives two answers, one positive and one negative.

记住，取平方根总是会得到两个答案，一个正一个负。

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6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant, often written as Δ. It tells us how many real roots a quadratic equation has, without solving it completely.

表达式 b² − 4ac 称为判别式，通常写作 Δ。它告诉我们二次方程有多少个实数根，而无需完全求解。

Δ = b² − 4ac

Discriminant | 判别式	Number of Real Roots | 实数根个数	Graph | 图像
Δ > 0	Two distinct real roots | 两个不相等的实数根	Crosses the x-axis twice | 与x轴有两个交点
Δ = 0	One repeated real root | 一个重根	Touches the x-axis once | 与x轴相切
Δ < 0	No real roots | 没有实数根	Does not meet the x-axis | 与x轴无交点

For example, the equation x² + 2x + 5 = 0 has Δ = 4 − 20 = −16, so it has no real roots.

例如，方程 x² + 2x + 5 = 0 的判别式 Δ = 4 − 20 = −16，因此它没有实数根。

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7. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, there are two useful relationships.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有两个有用的关系。

Sum of roots: α + β = −b/a

Product of roots: αβ = c/a

These relationships allow you to check your answers or to build a quadratic equation from given roots.

这些关系允许你检查答案，或从已知根构造二次方程。

Example | 例子: A quadratic equation has roots 2 and −3. Find the equation.

• Sum = 2 + (−3) = −1, so −b/a = −1.
• 和 = 2 + (−3) = −1，所以 −b/a = −1。
• Product = 2 × (−3) = −6, so c/a = −6.
• 积 = 2 × (−3) = −6，所以 c/a = −6。
• Taking a = 1 gives b = 1 and c = −6.
• 取 a = 1，得到 b = 1，c = −6。
• The equation is x² + x − 6 = 0.
• 方程为 x² + x − 6 = 0。

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8. Word Problems Involving Quadratics | 二次方程应用题

Quadratics are often used to model real-life situations such as areas, projectile motion and number problems. The key step is to translate the words into an equation, then solve it.

二次方程常用于模拟现实情境，如面积、抛体运动和数字问题。关键步骤是将文字转化为方程，然后求解。

Example | 例子: A rectangle is 3 cm longer than it is wide. Its area is 40 cm². Find its width.

• Let the width be x, so the length is x + 3.
• 设宽为 x，则长为 x + 3。
• Area equation: x(x + 3) = 40.
• 面积方程：x(x + 3) = 40。
• Expand: x² + 3x = 40, so x² + 3x − 40 = 0.
• 展开：x² + 3x = 40，所以 x² + 3x − 40 = 0。
• Factorise: (x + 8)(x − 5) = 0.
• 因式分解：(x + 8)(x − 5) = 0。
• Thus x = −8 or x = 5. Since width cannot be negative, the width is 5 cm.
• 因此 x = −8 或 x = 5。由于宽度不能为负，宽度为5厘米。

Always reject negative answers when the context requires a positive quantity.

当题目情境要求正数时，务必舍去负数解。

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

The graph of y = ax² + bx + c is a parabola. When a is positive, the curve opens upwards; when a is negative, it opens downwards.

y = ax² + bx + c 的图像是抛物线。当 a 为正时，曲线开口向上；当 a 为负时，开口向下。

• The x-intercepts are the roots of the equation ax² + bx + c = 0.
• 与x轴的交点是方程 ax² + bx + c = 0 的根。
• The y-intercept is the value of c.
• 与y轴的交点是 c 的值。
• The axis of symmetry is the vertical line x = −b/(2a).
• 对称轴是垂直线 x = −b/(2a)。
• The vertex lies on the axis of symmetry.
• 顶点位于对称轴上。

For example, y = x² − 4x + 3 has roots x = 1 and x = 3, a y-intercept of 3, and its axis of symmetry is x = 2.

例如，y = x² − 4x + 3 的根为 x = 1 和 x = 3，与y轴交点为3，对称轴为 x = 2。

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10. Intersection of a Line and a Quadratic | 直线与二次曲线的交点

To find where a straight line y = mx + k meets a parabola y = ax² + bx + c, substitute the line equation into the parabola equation and solve the resulting quadratic.

要求直线 y = mx + k 与抛物线 y = ax² + bx + c 的交点，将直线方程代入抛物线方程，然后解所得的二次方程。

ax² + bx + c = mx + k → ax² + (b − m)x + (c − k) = 0

• If the discriminant is positive, the line cuts the curve at two points.
• 若判别式为正，直线与曲线相交于两点。
• If the discriminant is zero, the line touches the curve at one point.
• 若判别式为零，直线与曲线相切于一点。
• If the discriminant is negative, the line does not meet the curve.
• 若判别式为负，直线与曲线没有交点。

These ideas also connect to the tangent condition in coordinate geometry, which is frequently tested.

这些想法还与坐标几何中的切线条件相关，这是常考的考点。

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11. Common Exam Mistakes | 常见考试错误

Many students lose marks through small but avoidable errors. Watch out for the following.

许多学生因为细小但可避免的错误而失分。请注意以下几点。

• Forgetting to rearrange the equation to the form ax² + bx + c = 0 before factorising.
• 在因式分解前忘记将方程整理为 ax² + bx + c = 0 的形式。
• Making sign errors when substituting negative values into the quadratic formula.
• 将负值代入求根公式时出现符号错误。
• Dividing both sides of the equation by x, which loses the solution x = 0.
• 方程两边同时除以x，从而丢失了 x = 0 这个解。
• Forgetting the ± sign when taking a square root.
• 取平方根时忘记 ± 号。
• Taking the square root of a negative number when the quadratic has no real roots.
• 当二次方程没有实数根时，对负数取平方根。

Write every step clearly and check your solutions by substituting them back into the original equation.

清晰地写出每一步，并通过将解代回原方程来检验答案。

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12. Practice Questions | 练习

Try these questions yourself. Solutions are given, but attempt them before checking.

请自己尝试以下题目。答案已给出，但请先尝试再核对。

Question | 题目	Answer | 答案
Solve x² − 7x + 10 = 0 | 解方程 x² − 7x + 10 = 0	x = 2 or x = 5 | x = 2 或 x = 5
Solve 3x² + 5x − 2 = 0 | 解方程 3x² + 5x − 2 = 0	x = 1/3 or x = −2 | x = 1/3 或 x = −2
Find the discriminant of x² + 6x + 9 = 0 | 求 x² + 6x + 9 = 0 的判别式	Δ = 0, one repeated root | Δ = 0，一个重根
A number squared plus 5 times the number equals 24. Find the positive number. | 一个数的平方加上这个数的5倍等于24。求这个正数。	x = 3 | x = 3

For the last question, note that x² + 5x = 24 becomes (x + 8)(x − 3) = 0, giving x = −8 or x = 3. Only x = 3 is positive.

对于最后一题，x² + 5x = 24 变为 (x + 8)(x − 3) = 0，得到 x = −8 或 x = 3。只有 x = 3 是正数。

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Quadratic equations are a bridge between algebra and geometry. Master the factorisation method, the quadratic formula, and the discriminant, and you will be well prepared for both calculator and non-calculator papers.

二次方程是代数与几何之间的桥梁。掌握因式分解法、求根公式和判别式，你就能为非计算器和计算器试卷做好充分准备。

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