Solving Quadratic Equations and Graphs | 二次方程与图像

📚 Solving Quadratic Equations and Graphs | 二次方程与图像

Quadratic equations appear in many areas of GCSE and IGCSE mathematics, from algebra to coordinate geometry. Mastering the methods of solving them and understanding their graphs is essential for exam success.

二次方程在 GCSE 和 IGCSE 数学中广泛出现，从代数到坐标几何都有涉及。掌握其解法并理解对应图像，是获得高分的关键。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants, with a ≠ 0. The highest power of the variable x is 2.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2。

Examples of quadratic equations include:

二次方程的例子包括：

• x² − 5x + 6 = 0
• 2x² + 3x − 4 = 0
• x² − 9 = 0

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2. The Standard Form and the Discriminant | 标准形式与判别式

Every quadratic equation can be written in the standard form ax² + bx + c = 0. The values of a, b and c determine the shape and position of its graph.

任何二次方程都可写成标准形式 ax² + bx + c = 0。a、b、c 的值决定了图像的形状与位置。

For the equation ax² + bx + c = 0, the discriminant is defined as:

对于方程 ax² + bx + c = 0，判别式定义为：

Δ = b² − 4ac

The discriminant tells us the nature of the roots:

判别式告诉我们根的性质：

Δ value | Δ 的值	Roots | 根
Δ > 0	Two distinct real roots | 两个不等实根
Δ = 0	One repeated real root | 一个二重实根
Δ < 0	No real roots | 没有实根

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3. Solving by Factorisation | 因式分解法

If a quadratic expression can be factorised into two linear factors, we can solve the equation by setting each factor equal to zero.

如果二次式可以分解为两个一次因式，则令每个因式等于零即可求解。

Example: Solve x² − 5x + 6 = 0.

例：解 x² − 5x + 6 = 0。

Factorise: (x − 2)(x − 3) = 0

分解因式：(x − 2)(x − 3) = 0

Therefore: x = 2 or x = 3.

因此：x = 2 或 x = 3。

This method works only if the expression is factorisable with simple integer factors.

该方法仅在表达式能用简单的整数因式分解时有效。

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4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form (x + p)² + q. This method works for any quadratic equation.

配方是将二次式改写成 (x + p)² + q 的形式。该方法适用于任何二次方程。

Example: Solve x² + 6x − 7 = 0 by completing the square.

例：用配方法解 x² + 6x − 7 = 0。

Complete the square: (x + 3)² − 9 − 7 = 0

配方：(x + 3)² − 9 − 7 = 0

(x + 3)² = 16

So x + 3 = ±4, giving x = 1 or x = −7.

所以 x + 3 = ±4，得到 x = 1 或 x = −7。

For a general expression ax² + bx + c, the completed square form is:

对于一般式子 ax² + bx + c，配方形式为：

a(x + b/(2a))² + (c − b²/(4a))

Notice that this formula uses only basic arithmetic, no division by zero issues when a ≠ 0.

注意该公式只用到基本算术，当 a ≠ 0 时不存在除以零的问题。

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5. The Quadratic Formula | 二次求根公式

The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0. It works for all real and even complex roots.

二次求根公式通过对一般方程 ax² + bx + c = 0 配方推导得出。它适用于所有实根甚至复根。

x = (−b ± √(b² − 4ac)) / (2a)

Remember: the expression under the square root is the discriminant Δ.

记住：根号下的表达式就是判别式 Δ。

Example: Solve 2x² − 4x − 3 = 0 using the quadratic formula.

例：用求根公式解 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substitute into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = 1 ± √10 / 2

So the two roots are 1 + √10/2 and 1 − √10/2.

因此两个根为 1 + √10/2 和 1 − √10/2。

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6. The Graph of y = ax² + bx + c | y = ax² + bx + c 的图像

The graph of a quadratic function is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

二次函数的图像是抛物线。当 a > 0 时，抛物线开口向上；当 a < 0 时，抛物线开口向下。

The y-intercept is the point where the graph crosses the y-axis, given by (0, c).

图像与 y 轴的交点为 (0, c)，称为 y 截距。

The x-intercepts are the roots of the equation ax² + bx + c = 0, if real roots exist.

图像与 x 轴的交点就是方程 ax² + bx + c = 0 的根（如果存在实根）。

The graph is symmetrical about a vertical line through its vertex.

图像关于通过顶点的竖直直线对称。

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7. Roots, Turning Points and Symmetry | 根、顶点与对称性

The turning point (also called the vertex) of a parabola is the maximum or minimum point. Its x-coordinate is always:

抛物线的顶点（也称转向点）是最大值点或最小值点。其 x 坐标为：

x = −b/(2a)

Once found, substitute this x-value back into the equation to find the y-coordinate.

求出 x 值后，将其代回原方程即可得到 y 坐标。

For example, for y = x² − 4x + 3, we have a = 1, b = −4, so x = 4/2 = 2. Then y = 4 − 8 + 3 = −1. The vertex is (2, −1).

例如，对于 y = x² − 4x + 3，a = 1，b = −4，所以 x = 4/2 = 2。于是 y = 4 − 8 + 3 = −1。顶点为 (2, −1)。

The axis of symmetry is the vertical line x = −b/(2a).

对称轴是竖直直线 x = −b/(2a)。

If the roots are p and q, the x-coordinate of the vertex is the midpoint: (p + q)/2.

如果根为 p 和 q，则顶点的 x 坐标是它们的中点：(p + q)/2。

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8. Sketching Quadratic Graphs | 二次函数图像的画法

To sketch a quadratic graph, follow these steps:

要画二次函数图像，请按以下步骤：

1. Identify the direction of opening (look at a).
2. 确定开口方向（看 a 的符号）。
3. Find the y-intercept (0, c).
4. 找到 y 截距 (0, c)。
5. Find the roots by solving ax² + bx + c = 0, if possible.
6. 通过解 ax² + bx + c = 0 求出根（如果可能）。
7. Find the vertex using x = −b/(2a).
8. 用 x = −b/(2a) 求出顶点。
9. Plot these points and draw a smooth parabola through them.
10. 标出这些点，并用平滑曲线画出抛物线。

If there are no real roots, the graph does not cross the x-axis; it lies entirely above or below it.

如果没有实根，图像不与 x 轴相交；它完全位于 x 轴上方或下方。

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9. Applications and Word Problems | 应用与应用题

Quadratic equations model many real-world situations, such as projectile motion, area problems and profit optimisation.

二次方程可模拟许多现实情境，例如抛体运动、面积问题及利润优化。

Example: A rectangle has length (x + 3) cm and width x cm. Its area is 28 cm². Find x.

例：一个长方形的长为 (x + 3) 厘米，宽为 x 厘米，面积为 28 平方厘米。求 x。

Set up the equation: x(x + 3) = 28

建立方程：x(x + 3) = 28

x² + 3x − 28 = 0

Factorise: (x + 7)(x − 4) = 0

分解因式：(x + 7)(x − 4) = 0

So x = −7 or x = 4. Since a length cannot be negative, x = 4.

因此 x = −7 或 x = 4。由于长度不能为负，x = 4。

Always check whether your solutions make sense in the original context.

务必检查解在原始情境中是否有意义。

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10. Common Mistakes and Revision Tips | 常见错误与复习建议

Many students make the same mistakes when working with quadratics. Avoiding them can raise your grade significantly.

许多学生在做二次方程时会犯相同错误。避免这些错误能显著提高成绩。

• Forgetting to set the equation to zero before factorising.
• 因式分解前忘记将方程化为零。
• Losing the ± sign when taking square roots.
• 开平方时遗漏 ± 号。
• Using the quadratic formula with incorrect values of a, b, c.
• 使用求根公式时 a、b、c 的取值错误。
• Ignoring the discriminant when deciding whether roots exist.
• 判断根是否存在时忽略判别式。
• Dropping negative signs when completing the square.
• 配方时丢掉负号。

To revise effectively, practise all three methods of solving, and always sketch graphs of your answers to check consistency.

为了高效复习，请练习三种解法，并务必画出图像来检查答案是否一致。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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