📚 Using Graphs to Solve Quadratic Equations | 用图解法解二次方程
raphs of quadratic functions to find solutions to equations of the form ax² + bx + c = 0. You will learn to plot parabolas, identify x-intercepts (roots), and even use intersections with straight lines to solve more complex problems. Mastering this skill is essential for your Edexcel IGCSE exam and helps you connect algebra to geometry.
二次方程是IGCSE数学的基础内容。尽管因式分解和公式法等代数方法很常用，但通过图像求解二次方程是一种强大的可视化方法，能加深对概念的理解。本文将解释如何利用二次函数图像来求解形如 ax² + bx + c = 0 的方程。你将学习绘制抛物线、识别与 x 轴的交点（根），甚至利用与直线的交点来解决更复杂的问题。掌握这项技能对 Edexcel IGCSE 考试至关重要，也能帮助你建立代数与几何之间的联系。
1. What is a Quadratic Graph? | 什么是二次图像？
onstants and a ≠ 0. Its graph is a smooth, U-shaped curve called a parabola. If a > 0, the parabola opens upwards (positive quadratic), and if a < 0, it opens downwards (negative quadratic). The shape is symmetrical, and the turning point is called the vertex. Understanding these basic features is the first step towards using graphs to solve equations.
二次函数的一般形式为 y = ax² + bx + c，其中 a、b 和 c 为常数，且 a ≠ 0。它的图像是一条光滑的 U 形曲线，称为抛物线。如果 a > 0，抛物线开口向上（正二次函数）；如果 a < 0，则开口向下（负二次函数）。图形是对称的，其转折点称为顶点。理解这些基本特征是使用图像求解方程的第一步。
2. Connecting the Equation to the Graph | 方程与图像的关联
he values of x that make the quadratic expression equal to zero. If the graph touches the x-axis at one point, the equation has one repeated root. If it does not intersect the x-axis, there are no real solutions.
要通过图像求解二次方程，例如 ax² + bx + c = 0，我们需要考虑函数 y = ax² + bx + c。方程的解（根）对应于图像与 x 轴交点的 x 坐标，即 y = 0 的位置。也就是说，x 轴截距就是使二次表达式等于零的 x 值。如果图像与 x 轴相切于一个点，则方程有一个重根。如果图像不与 x 轴相交，则无实数解。
3. Key Features to Identify on a Parabola | 抛物线的关键特征
intercept (where x = 0, so y = c); the roots (x-intercepts); the line of symmetry given by x = −b/(2a); and the vertex (turning point). The vertex is either a minimum or maximum and lies on the line of symmetry. By plotting these features accurately, you can draw a reliable curve to read off solutions. A table of values often helps when the algebraic form is not easily factorised.
在绘制或描点画二次图像时，应识别以下特征：y 轴截距（当 x = 0 时，y = c）；根（x 轴截距）；对称轴为 x = −b/(2a)；以及顶点（转折点）。顶点可以是最低点或最高点，并位于对称轴上。通过精确标出这些特征，可以绘制出可靠的曲线以读取解。当二次式不易因式分解时，数值表格往往很有帮助。
4. Step-by-Step: Solving by Plotting the Quadratic | 分步指导：通过描点画图求解
ts. The solutions are the x-coordinates where the curve cuts the x-axis. For this example, the graph crosses at x = −1 and x = 5, so the roots are −1 and 5.
要用图像法求解如 x² − 4x − 5 = 0 的方程，首先为 y = x² − 4x − 5 制作数值表格。选择 x 值，最好包含负数和正数，并计算对应的 y 值。然后在坐标纸上描点，确保刻度平滑、均匀。通过点画出最佳拟合的抛物线。方程的解就是曲线与 x 轴相交处的 x 坐标。在这个例子中，图像与 x 轴交于 x = −1 和 x = 5，因此根为 −1 和 5。
5. Drawing an Accurate Parabola Using Symmetry | 利用对称性绘制准确的抛物线
the line of symmetry to find mirror points. For y = ax² + bx + c, the axis of symmetry is x = −b/(2a). After calculating the vertex, choose a few x-values on one side, compute y, and reflect them across the axis to get points on the other side. This technique saves time and increases accuracy, which is crucial when solutions are not integers and must be estimated from the graph.
你无需绘制大量点。可以利用对称轴来寻找对称点。对于 y = ax² + bx + c，对称轴为 x = −b/(2a)。计算出顶点后，在对称轴一侧选取几个 x 值并计算 y，然后将它们对称反射到另一侧以得到对应的点。这种方法节省时间并提高准确性，尤其当解不是整数、需要从图像上估算时，这一点至关重要。
6. Solving When the Graph Does Not Factorise Neatly | 当图像无法顺利因式分解时的求解
an read approximate roots from the x-intercepts. The graph will cross between two integer x-values, allowing you to estimate roots to one decimal place, which is often required in IGCSE papers. Always label the scale clearly and draw a smooth curve.
许多考题涉及的二次方程无法用简单的整数分解。例如，求解 x² + 2x − 2 = 0。配方法或求根公式会得到带根号的答案。但通过绘制 y = x² + 2x − 2 的图像，你可以从 x 轴截距读取近似根。图像会在两个整数 x 值之间穿过 x 轴，从而估算出精确到一位小数的根，这在 IGCSE 试卷中经常要求。务必清晰标注坐标刻度并画出平滑曲线。
7. The Role of the Discriminant in Graphical Solutions | 判别式在图解法中的作用
The discriminant Δ = b² − 4ac x touches the x-axis, giving one repeated root. If Δ < 0, the parabola does not meet the x-axis, meaning the quadratic equation has no real solutions. Before starting your graph, calculate Δ to predict what you should see – it helps you check whether your sketch is correct.
判别式 Δ = b² − 4ac 决定了抛物线与 x 轴相交的次数。若 Δ > 0，图像与 x 轴有两个不同的交点，即有两个实数根。若 Δ = 0，顶点与 x 轴相切，即有一个重根。若 Δ < 0，抛物线与 x 轴不相交，这意味着二次方程无实数解。在动手绘图前，先计算判别式可以预测图像的交点情况——这有助于检查你的草图是否正确。
8. Solving Quadratic Equations Using Intersections with a Line | 利用与直线的交点解二次方程
axes. The x-coordinates of the intersections between y = x² and the line are the solutions
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