Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is a polynomial equation of degree 2. Its general form is ax² + bx + c = 0, where a, b, c are constants and a ≠ 0. Solving quadratic equations is a fundamental skill in IGCSE mathematics, as it appears in algebra, geometry and many real-world problems.

二次方程是最高次数为 2 的多项式方程，其一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。解二次方程是 IGCSE 数学中的基本技能，它出现在代数、几何以及许多实际问题中。

1. Standard Form | 标准形式

A quadratic equation must be rearranged into the standard form ax² + bx + c = 0 before solving. This allows us to compare coefficients and apply the most appropriate method. If the equation is not given in this form, expand brackets and collect like terms first.

解二次方程前，必须先整理成标准形式 ax² + bx + c = 0。这样便于比较系数，并选用最合适的方法。如果方程不是这种形式，要先展开括号并合并同类项。

For example, x² = 4x + 5 must be rewritten as x² − 4x − 5 = 0.

例如，x² = 4x + 5 必须改写为 x² − 4x − 5 = 0。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic can be written as a product of two linear factors. Set each factor equal to zero and solve the resulting linear equations.

当二次式能写成两个一次因式的乘积时，因式分解往往是最快捷的方法。令每个因式等于零，再解所得的一次方程。

Solve x² + 5x + 6 = 0. Since x² + 5x + 6 = (x + 2)(x + 3), we obtain x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.

解方程 x² + 5x + 6 = 0。因为 x² + 5x + 6 = (x + 2)(x + 3)，所以 x + 2 = 0 或 x + 3 = 0，即 x = −2 或 x = −3。

Always expand your factors to check the original expression.

务必展开因式以检验是否等于原表达式。

3. Solving by Quadratic Formula | 公式法

The quadratic formula works for any quadratic equation, including those that cannot be factorised easily. When the equation is in standard form, the roots are given by the formula below.

公式法适用于任何二次方程，包括那些不易因式分解的方程。当方程为标准形式时，根由下面的公式给出。

x = (−b ± √(b² − 4ac)) / 2a

Here the symbol ± means that we add or subtract the square root to obtain two solutions.

这里的符号 ± 表示加上或减去根号中的值，从而得到两个解。

For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2. Substitute into the formula to get x = 0.5 or x = −2.

例如，解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入公式计算，得 x = 0.5 或 x = −2。

4. Solving by Completing the Square | 配方法

Completing the square transforms x² + bx + c into (x + p)² + q. This method is useful for solving equations and also helps in sketching graphs.

配方法将 x² + bx + c 变形为 (x + p)² + q。这种方法既可用于解方程，也有助于画图象。

To solve x² + 6x + 4 = 0, write (x + 3)² − 9 + 4 = 0, so (x + 3)² = 5, then x = −3 ± √5.

解 x² + 6x + 4 = 0 时，将其写成 (x + 3)² − 9 + 4 = 0，即 (x + 3)² = 5，于是 x = −3 ± √5。

Remember to divide by a first if the coefficient of x² is not 1.

如果 x² 的系数不是 1，记得先将二次项系数化为 1。

5. The Discriminant | 判别式

The discriminant is the expression Δ = b² − 4ac inside the quadratic formula. Its value determines the nature of the roots without solving the equation.

判别式是公式法中被开方的部分 Δ = b² − 4ac。它无需解方程即可判断根的性质。

Discriminant	Nature of Roots
Δ > 0	Two distinct real roots
Δ = 0	One repeated real root
Δ < 0	No real roots (two complex roots)

对应中文：Δ > 0 时有两个不同的实数根；Δ = 0 时有一个重根；Δ < 0 时没有实数根（有两个复数根）。

6. Roots of a Quadratic Equation | 方程的根

If α and β are the roots of ax² + bx + c = 0, then their sum and product are related to the coefficients. This is useful for checking answers and for constructing equations from given roots.

若 α 和 β 是二次方程 ax² + bx + c = 0 的两个根，则根的和与积与系数有关。这个关系可用于检验答案，也可由已知的根构造方程。

α + β = −b/a, αβ = c/a

For the equation x² − 5x + 6 = 0, the roots are 2 and 3, so their sum is 5 and product is 6.

方程 x² − 5x + 6 = 0 的根为 2 和 3，所以它们的和为 5，积为 6。

7. Graphs of Quadratic Functions | 二次函数图象

The graph of y = ax² + bx + c is a parabola. When a > 0 it opens upward; when a < 0 it opens downward. The x-intercepts are the real roots of the equation, and the vertex lies on the axis of symmetry x = −b/(2a).

函数 y = ax² + bx + c 的图象是抛物线。当 a > 0 时开口向上；当 a < 0 时开口向下。图象与 x 轴的交点就是方程的实数根，顶点位于对称轴 x = −b/(2a) 上。

Sketching a quadratic curve helps relate algebra to geometry, especially when solving inequalities.

画二次函数草图有助于将代数与几何联系起来，特别在解不等式时非常有效。

8. Word Problems | 应用题

Many real-life situations involving area, motion or profit can be modelled by quadratic equations. Set up the equation carefully from the given information, then solve and check the validity of your answers.

许多实际问题，如面积、运动或利润问题，都可以用二次方程建模。根据题意认真建立方程，解出后还要检验根的合理性。

For example, a rectangle has length 3 cm longer than its width and area 40 cm². Let width = x, so length = x + 3. Then x(x + 3) = 40, giving x² + 3x − 40 = 0. Solve to get x = 5 (the negative root is rejected).

例如：一个长方形的长比宽多 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3。于是 x(x + 3) = 40，即 x² + 3x − 40 = 0。解得 x = 5（负根舍去）。

9. Common Mistakes | 常见错误

Students often forget to set the equation to zero before factorising. Another common error is losing a negative sign when applying the quadratic formula. Also, when extracting a square root, remember the ± symbol.

学生经常忘记在因式分解前将方程整理为零。另一个常见错误是在套用公式法时丢掉负号。此外，开平方时不要忘记取 ±。

Always simplify the final answer to exact values or to the required degree of accuracy.
通常将最终答案化为精确值，或按题目要求保留相应精度。
Check each solution by substituting back into the original equation.
将每个解代入原方程进行检验。

10. Practice Questions | 练习题

Try the following problems, using any appropriate method. Keep your working clear and show every step.

请练习以下题目，选用合适的方法，过程要清晰并写出每一步。

1. x² − 7x + 12 = 0	2. 2x² + 5x − 3 = 0
3. x² = 9x	4. 3x² + 6x + 2 = 0

Answers: 1. x = 3 or 4; 2. x = 0.5 or −3; 3. x = 0 or 9; 4. x = −1 ± √3/3.

答案：1. x = 3 或 4；2. x = 0.5 或 −3；3. x = 0 或 9；4. x = −1 ± √3/3。

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Solving Quadratic Equations | 解二次方程