📚 Solving Quadratic Equations | 解二次方程
his revision guide, we will explore the key methods for solving quadratic equations, understand the discriminant, and apply these skills to real problems.
二次方程是形如 ax² + bx + c = 0 的方程，其中 a ≠ 0。在本复习指南中，我们将探讨解二次方程的主要方法，理解判别式，并将这些技能应用到实际问题中。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation always has the highest power of x equal to 2. The general form is:
二次方程的最高次幂始终为 2。其一般形式为：
ax² + bx + c = 0
comes linear. The values of x that satisfy the equation are called the roots or solutions.
这里，a、b 和 c 是常数，且 a ≠ 0。如果 a = 0，方程就变成了一次方程。满足方程的 x 值称为方程的根或解。
For example, rearrange 3x² - 4x = 5 into standard form by subtracting 5 from both sides:
例如，将 3x² - 4x = 5 移项为标准形式：两边同时减去 5：
3x² - 4x - 5 = 0
2. Solving by Factorisation | 因式分解法
the zero product property. If (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0.
如果二次方程可以写成两个一次因式的乘积，我们可以利用零乘积性质。若 (px + q)(rx + s) = 0，则要么 px + q = 0，要么 rx + s = 0。
Solve x² - 5x + 6 = 0 by factorisation.
用因式分解法解 x² - 5x + 6 = 0。
We look for two numbers that multiply to 6 and add to -5. These are -2 and -3.
我们需要找到两个数，它们相乘等于 6，相加等于 -5。这两个数是 -2 和 -3。
(x - 2)(x - 3) = 0
Therefore x = 2 or x = 3.
因此 x = 2 或 x = 3。
Always check your answers by substituting back into the original equation.
始终将答案代回原方程进行检验。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful when factorisation is difficult or when finding turning points.
配方法将二次式改写为 a(x + p)² + q 的形式。这在因式分解困难或需要求顶点时非常有用。
der x² + 6x + 4 = 0. First, halve the coefficient of x: 6 ÷ 2 = 3. Then square it: 3² = 9.
考虑 x² + 6x + 4 = 0。首先，取 x 系数的一半：6 ÷ 2 = 3。然后平方：3² = 9。
Rewrite the equation as:
将方程改写为：
(x + 3)² - 9 + 4 = 0
(x + 3)² = 5
Take the square root of both sides, remembering the ± sign:
两边开平方，注意 ± 号：
x + 3 = ±√5
x = -3 ± √5
4. The Quadratic Formula | 二次求根公式
olves any quadratic equation, even when factorisation is impossible. For ax² + bx + c = 0:
二次求根公式可以解任何二次方程，即使无法因式分解。对于 ax² + bx + c = 0：
x = (-b ± √b² - 4ac) / (2a)
Use this formula with care. Substitute the values of a, b and c exactly as they appear.
使用此公式时要小心。将 a、b 和 c 的准确值代入。
Solve 2x² + 3x - 5 = 0 using the quadratic formula. Here, a = 2, b = 3, c = -5.
用二次公式解 2x² + 3x - 5 = 0。这里 a = 2，b = 3，c = -5。
x = (-3 ± √(3² - 4 × 2 × (-5))) / (2 × 2)
x = (-3 ± √9 + 40) / 4 = (-3 ± √49) / 4 = (-3 ± 7) / 4
So x = (4) / 4 = 1 or x = (-10) / 4 = -2.5.
因此 x = 4 / 4 = 1 或 x = -10 / 4 = -2.5。
5. The Discriminant b² - 4ac | 判别式 b² - 4ac
ant determines the number and type of roots of a quadratic equation. For ax² + bx + c = 0:
判别式决定二次方程的根的个数和类型。对于 ax² + bx + c = 0：
| Value of b² - 4ac | Number of real roots | Graphical meaning |
| b² - 4ac > 0 | Two distinct real roots | The curve crosses the x-axis at two points |
| b² - 4ac = 0 | One repeated real root | The curve touches the x-axis at one point |
| b² - 4ac < 0 | No real roots | The curve does not meet the x-axis |
the discriminant is (-4)² - 4 × 1 × 4 = 16 - 16 = 0, so there is exactly one root (x = 2).
例如，在 x² - 4x + 4 = 0 中，判别式为 (-4)² - 4 × 1 × 4 = 16 - 16 = 0，因此恰好有一个根（x = 2）。
6. Quadratic Graphs and Roots | 二次函数图像与根
0 are the x-coordinates where the graph of y = ax² + bx + c crosses or touches the x-axis.
二次方程 ax² + bx + c = 0 的解是函数 y = ax² + bx + c 的图像与 x 轴相交或相切处的 x 坐标。
If a > 0, the parabola opens upwards; if a < 0, it opens downwards. The turning point can be found by completing the square.
若 a > 0，抛物线开口向上；若 a < 0，开口向下。可以通过配方法找到顶点。
1. The vertex is at (2, -1), and the roots are x = 1 and x = 3 because (x - 1)(x - 3) = 0.
对于 y = x² - 4x + 3，配平方得 (x - 2)² - 1。顶点为 (2, -1)，根为 x = 1 和 x = 3，因为 (x - 1)(x - 3) = 0。
Sketch the graph by marking the roots, the vertex, and the y-intercept (where x = 0).
画草图时，标出根、顶点和 y 截距（x = 0 处的值）。
7. Problem Solving with Quadratics | 二次方程应用题
Many practical problems lead to quadratic equations. Read the problem carefully, define a variable, form an equation, and solve it.
许多实际问题会导出二次方程。仔细读题，设变量，列方程，再求解。
Example: The length of a rectangle is 4 cm more than its width. Its area is 45 cm². Find the width.
例：一个长方形的长比宽多 4 cm，面积为 45 cm²。求宽。
Let the width be w cm. Then the length is (w + 4) cm. Area = w(w + 4) = 45.
设宽为 w cm，则长为 (w + 4) cm。面积 = w(w + 4) = 45。
w² + 4w - 45 = 0
+ 9)(w - 5) = 0, so w = -9 or w = 5. Since a length cannot be negative, the width is 5 cm.
因式分解：(w + 9)(w - 5) = 0，所以 w = -9 或 w = 5。因为长度不能为负，所以宽为 5 cm。
Always reject solutions that do not make sense in the context.
始终舍去在情境中不合理的解。
8. Common Mistakes and Tips | 常见错误与提示
to zero before applying methods. Also, when taking square roots, do not forget the ± sign.
学生在应用方法之前常常忘记将方程移项为零。此外，开平方时不要忘记 ± 号。
Always collect like terms and write the equation in the form ax² + bx + c = 0.
始终合并同类项，并将方程写成 ax² + bx + c = 0 的形式。
When factorising, check the signs carefully. For example, x² - 7x + 12 = (x - 3)(x - 4).
因式分解时仔细检查符号。例如，x² - 7x + 12 = (x - 3)(x - 4)。
Use the quadratic formula when told to give answers to a certain number of decimal places or significant figures.
当题目要求答案保留指定小数位或有效数字时，使用二次求根公式。
For word problems, check that your final answer satisfies the original condition.
对于文字题，检查最终答案是否满足原题条件。
Practise all three methods: factorisation, completing the square, and the quadratic formula. Being fluent in each method gives you flexibility in the exam.
练习所有三种方法：因式分解、配方和二次公式。熟练掌握每种方法能让你在考试中灵活应对。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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