📚 Quadratic Equations and Graphs | 二次方程与图像
Quadratic equations are a central topic in IGCSE Mathematics, appearing in algebra, coordinate geometry and modelling. A quadratic equation in one variable has the form ax² + bx + c = 0 with a ≠ 0. This article covers standard methods of solving quadratics, the discriminant, graphs, inequalities and typical applications.
二次方程是 IGCSE 数学的核心主题，出现在代数、坐标几何和建模中。一元二次方程的形式为 ax² + bx + c = 0，其中 a ≠ 0。本文介绍二次方程的标准解法、判别式、图像、不等式以及典型应用。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation in one variable is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The condition a ≠ 0 is essential because if a = 0 the equation becomes linear. The expression ax² + bx + c is called a quadratic expression.
一元二次方程是任何可以写成 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。条件 a ≠ 0 非常重要，因为如果 a = 0，方程就变成了一次方程。表达式 ax² + bx + c 称为二次三项式。
Before solving, always rearrange the equation so that one side is zero. For example, 3x² = 5x + 2 becomes 3x² − 5x − 2 = 0. This standard form makes it easier to identify a, b and c correctly.
解题前，一定要把方程整理成一边为零的形式。例如 3x² = 5x + 2 要化为 3x² − 5x − 2 = 0。标准形式有助于正确识别 a、b 和 c。
2. Solving by Factorising | 因式分解法解二次方程
When a quadratic expression can be factorised, the zero product property gives a quick method. For x² + 5x + 6 = 0, look for two numbers that multiply to 6 and add to 5; these are 2 and 3. Therefore (x + 2)(x + 3) = 0, giving x = −2 or x = −3.
当二次式可以因式分解时，零积性质提供了一种快速解法。对于 x² + 5x + 6 = 0，找两个数，它们相乘得 6、相加得 5；这两个数是 2 和 3。于是 (x + 2)(x + 3) = 0，得到 x = −2 或 x = −3。
If a ≠ 1, use trial and error or split the middle term. For 2x² + 7x + 3, rewrite 7x as 6x + x, then factor by grouping: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
如果 a ≠ 1，可以使用尝试法或拆中项法。例如 2x² + 7x + 3，把 7x 拆成 6x + x，再分组分解：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
3. Completing the Square | 配方法
Completing the square rewrites x² + bx as (x + p)² − q. For x² + 6x + 2 = 0, half of 6 is 3, so x² + 6x = (x + 3)² − 9. Hence the equation becomes (x + 3)² − 7 = 0, so (x + 3)² = 7 and x = −3 ± √7.
配方法把 x² + bx 改写为 (x + p)² − q。对于 x² + 6x + 2 = 0，6 的一半是 3，因此 x² + 6x = (x + 3)² − 9。于是方程变为 (x + 3)² − 7 = 0，所以 (x + 3)² = 7，解得 x = −3 ± √7。
When a is not 1, first divide through by a or factor out the leading coefficient before completing the square. For example, 2x² + 8x + 3 = 0 becomes 2(x² + 4x) + 3 = 0, then complete the square inside the bracket.
当 a 不等于 1 时，要先把方程两边除以 a 或把首项系数提到括号外，然后再配方。例如 2x² + 8x + 3 = 0 可写成 2(x² + 4x) + 3 = 0，再在括号内配方。
4. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0. It states:
求根公式可以解任意二次方程 ax² + bx + c = 0。公式为：
x = (−b ± √(b² − 4ac)) / (2a)
This formula is derived by completing the square on the general form. Example: for 2x² + 3x − 2 = 0, a = 2, b = 3, c = −2, so x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4, giving x = 0.5 or x = −2.
这个公式由一般形式配方推导而来。例：对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2，所以 x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4，得到 x = 0.5 或 x = −2。
Write the formula carefully and always substitute values with their signs. Use brackets when entering into a calculator to avoid sign errors.
书写公式时要仔细，代入数值时要保留正负号。使用计算器时建议加上括号，以避免符号错误。
5. The Discriminant | 判别式
The expression D = b² − 4ac is called the discriminant. If D > 0, there are two distinct real roots. If D = 0, there is one repeated real root. If D < 0, there are no real roots, but two complex roots.
表达式 D = b² − 4ac 称为判别式。如果 D > 0，方程有两个不相等的实数根；如果 D = 0，有一个重根；如果 D < 0，没有实数根，但有两个复数根。
For example, to find k such that x² + kx + 9 = 0 has equal roots, set D = k² − 36 = 0, giving k = ±6. This technique is very common in IGCSE exam questions.
例如，要使 x² + kx + 9 = 0 有重根，令 D = k² − 36 = 0，解得 k = ±6。这种技巧在 IGCSE 考试中非常常见。
6. Graphs of Quadratic Functions | 二次函数图像
The graph of y = ax² + bx + c is a parabola. Its axis of symmetry is the vertical line x = −b / (2a). The vertex lies on this line, and its y-coordinate is found by substituting x = −b / (2a) into the function. If a > 0 the parabola opens upward and the vertex is a minimum; if a < 0 it opens downward and the vertex is a maximum.
函数 y = ax² + bx + c 的图像是一条抛物线。它的对称轴是竖直线 x = −b / (2a)。顶点在对称轴上，顶点的纵坐标是将 x = −b / (2a) 代入函数得到的值。如果 a > 0，抛物线开口向上，顶点是最低点；如果 a < 0，开口向下，顶点是最高点。
The y-intercept is c, because y(0) = c. The x-intercepts are the real roots of ax² + bx + c = 0. Sketching requires marking the vertex, the y-intercept, and any real x-intercepts, then drawing a smooth symmetric curve.
y 轴截距是 c，因为 y(0) = c。x 轴截距就是方程 ax² + bx + c = 0 的实数根。画草图时需要标出顶点、y 轴截距和所有实数 x 轴截距，然后画出平滑对称的曲线。
7. Sum and Product of Roots | 根与系数的关系
If the roots of ax² + bx + c = 0 are α and β, then the sum α + β = −b/a and the product αβ = c/a. These are Vieta’s formulas and are useful for forming equations or checking answers.
如果方程 ax² + bx + c = 0 的两个根是 α 和 β，那么 α + β = −b/a，αβ = c/a。这称为韦达定理，常用于构造方程或检验答案。
Example: if roots are 2 and −3, then the equation is x² − (sum)x + product = x² − (−1)x + (−6) = x² + x − 6 = 0.
例：如果根是 2 和 −3，那么方程为 x² −（两根之和）x + 两根之积 = x² − (−1)x + (−6) = x² + x − 6 = 0。
8. Quadratic Inequalities | 二次不等式
To solve a quadratic inequality such as x² − 4x + 3 < 0, first solve the corresponding equation x² − 4x + 3 = 0 to get x = 1 and x = 3. These are the critical values. Test intervals or use the shape of the parabola. Since a > 0, the graph is below the x-axis between the roots, so the solution is 1 < x < 3.
解二次不等式（如 x² − 4x + 3 < 0）时，先解对应方程 x² − 4x + 3 = 0，得到 x = 1 和 x = 3，它们是临界值。然后检验区间或利用抛物线形状。因为 a > 0，图像在两个根之间位于 x 轴下方，所以解集为 1 < x < 3。
For ≥ or ≤, include the roots only when the inequality allows equality, and use closed intervals. For example, x² − 4x + 3 ≤ 0 has solution 1 ≤ x ≤ 3.
对于 ≥ 或 ≤，仅当不等式允许等号时才包含根，并使用闭区间。例如 x² − 4x + 3 ≤ 0 的解集为 1 ≤ x ≤ 3。
9. Applications and Word Problems | 应用与文字题
Quadratic equations appear in area problems, projectile motion, profit optimisation and number puzzles. For a rectangle of length 5 m longer than its width with area 24 m², let width be w. Then w(w + 5) = 24, so w² + 5w − 24 = 0, giving w = 3 or w = −8; reject the negative and conclude width 3 m, length
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导