Quadratic Equations: Methods and Graphs | 二次方程：解法与图像

📚 Quadratic Equations: Methods and Graphs | 二次方程：解法与图像

In IGCSE Mathematics, quadratic equations appear in algebra, coordinate geometry, and problem-solving. A quadratic equation is an equation of degree 2, usually written as ax² + bx + c = 0. The ability to solve quadratics by factorising, completing the square, and using the quadratic formula is essential, as is understanding the shape and key features of the graph y = ax² + bx + c.

在 IGCSE 数学中，二次方程出现在代数、坐标几何和应用题中。二次方程是二次的方程，通常写成 ax² + bx + c = 0。掌握因式分解法、配方法和公式法求解二次方程，以及理解图像 y = ax² + bx + c 的形状和关键特征，都是必备技能。

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

A quadratic equation is a polynomial equation of degree 2. It has the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is crucial because if a = 0 the equation becomes linear.

二次方程是二次的多项式方程。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。a ≠ 0 这一条件很关键，因为如果 a = 0，方程就变成了一次方程。

The highest power of the variable x is 2. Quadratic equations can have two real solutions, one repeated real solution, or no real solutions. Later you will use the discriminant to decide which case occurs.

变量 x 的最高次数为 2。二次方程可能有两个实数解、一个重根实数解，或者没有实数解。稍后你将使用判别式来判断属于哪种情况。

Common examples include x² – 5x + 6 = 0, 2x² + 3x – 2 = 0, and x² – 4 = 0. Equations such as x³ + x – 1 = 0 are not quadratic because the highest power is 3.

常见的例子包括 x² – 5x + 6 = 0、2x² + 3x – 2 = 0 和 x² – 4 = 0。像 x³ + x – 1 = 0 这样的方程不是二次方程，因为最高次数是 3。

2. Standard Form and Coefficients | 标准形式与系数

Before solving, rearrange the equation into standard form ax² + bx + c = 0. Collect all terms on one side so the other side is zero. This makes the values of a, b and c clear.

在求解之前，先把方程整理成标准形式 ax² + bx + c = 0。把所有项移到一边，使另一边为零。这样可以清楚地看出 a、b 和 c 的值。

For example, 3x – x² = 4 can be rearranged as -x² + 3x – 4 = 0, or equivalently x² – 3x + 4 = 0 after multiplying by -1. Identifying the correct signs is essential for factorising and using the formula.

例如，3x – x² = 4 可以整理为 -x² + 3x – 4 = 0，也可以乘以 -1 等价地写成 x² – 3x + 4 = 0。正确识别符号对于因式分解和公式法都很重要。

Always check whether the equation is already in standard form. An equation like x² = 9 is better written as x² – 9 = 0 before deciding on a method.

始终检查方程是否已经是标准形式。像 x² = 9 这样的方程，最好先写成 x² – 9 = 0，再决定使用哪种方法。

3. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic expression factorises nicely. The method relies on the zero product property: if p × q = 0, then p = 0 or q = 0.

当二次表达式容易因式分解时，因式分解法通常是最快的方法。该方法依赖于零乘积性质：如果 p × q = 0，那么 p = 0 或 q = 0。

To solve x² – 5x + 6 = 0, factorise the left side as (x – 2)(x – 3). Setting each factor equal to zero gives x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.

求解 x² – 5x + 6 = 0，将左边因式分解为 (x – 2)(x – 3)。令每个因式等于零，得到 x – 2 = 0 或 x – 3 = 0，所以 x = 2 或 x = 3。

For 2x² + 5x – 3 = 0, factorise as (2x – 1)(x + 3) = 0. This gives 2x – 1 = 0 or x + 3 = 0, so x = 1/2 or x = -3.

对于 2x² + 5x – 3 = 0，因式分解为 (2x – 1)(x + 3) = 0。由此得到 2x – 1 = 0 或 x + 3 = 0，所以 x = 1/2 或 x = -3。

After finding solutions, substitute them back into the original equation to check. This is a quick way to catch sign errors or factorising mistakes.

求出解后，将它们代回原方程进行检验。这是快速发现符号错误或因式分解错误的方法。

4. Solving by Completing the Square | 配方法

Completing the square transforms a quadratic expression into the form (x + p)² + q. This method is useful when the quadratic does not factorise easily and when you need to find the vertex of a parabola.

配方法把二次表达式转化为 (x + p)² + q 的形式。当二次式不容易因式分解，以及需要求抛物线顶点时，这个方法很有用。

To solve x² + 6x + 2 = 0, first write x² + 6x = -2. Half of 6 is 3, and 3² = 9, so add 9 to both sides: x² + 6x + 9 = 7. This gives (x + 3)² = 7.

求解 x² + 6x + 2 = 0，先写成 x² + 6x = -2。6 的一半是 3，3² = 9，所以两边加上 9：x² + 6x + 9 = 7。这给出 (x + 3)² = 7。

Taking square roots gives x + 3 = ±√7. Therefore x = -3 ± √7. Write the two solutions separately: x = -3 + √7 or x = -3 – √7.

取平方根得到 x + 3 = ±√7。因此 x = -3 ± √7。分别写出两个解：x = -3 + √7 或 x = -3 – √7。

When the coefficient of x² is not 1, divide every term by that coefficient first. For example, 2x² + 8x + 5 = 0 becomes x² + 4x + 5/2 = 0 before

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