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

A quadratic equation is a polynomial equation of degree two. In IGCSE Mathematics and A-Level algebra, you must solve quadratic equations by factorising, completing the square, and using the quadratic formula. You must also understand how the graph of a quadratic function links to its roots, axis of symmetry and vertex.

二次方程是次数为二的多项式方程。在 IGCSE 数学和 A-Level 代数中，你需要掌握用因式分解法、配方法和求根公式解二次方程，同时理解二次函数图像与根、对称轴和顶点的联系。

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

ants and a ≠ 0. The highest power of the unknown x is 2, which gives the equation its name 'quadratic' adratic equations often have two solutions, but they can also have one repeated solution or no real solutions. The name 'quadratic' comes from the Latin word 'quadratus', meaning square, because the variable is squared.

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数且 a ≠ 0。未知数 x 的最高次数是 2，因此称为二次方程。如果 a = 0，方程就变成一次方程而不是二次方程。系数 a 控制抛物线的弯曲程度，b 和 c 影响图像的位置。二次方程通常有两个解，但也可能有一个重根或没有实数解。'quadratic' 一词来源于拉丁语 'quadratus'，意为正方形，因为变量被平方。

ax² + bx + c = 0, a ≠ 0

= 0 and x² − 9 = 0. Each one has degree two after all terms are collected and simplified.

例如 x² − 5x + 6 = 0、3x² − 5x + 2 = 0 和 x² − 9 = 0。每一项在合并化简后，最高次数都是 2。

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

are real numbers, and a cannot be zero. Identifying these values correctly is essential for using the quadratic formula and the discriminant.

在求解之前，你应把方程整理成标准形式：ax² + bx + c = 0。系数 a、b、c 是实数，且 a 不能为零。正确识别这些值对使用求根公式和判别式至关重要。

x, you must first rearrange it to 2x² − 5x + 3 = 0, then identify a = 2, b = −5 and c = 3.

例如，方程 3x² − 5x + 2 = 0 中，a = 3、b = −5、c = 2。方程 2x² + 3 = 5x 必须先整理为 2x² − 5x + 3 = 0，然后再确定 a = 2、b = −5、c = 3。

Always simplify the equation first by collecting like terms and moving all terms to one side. Missing a negative sign is one of the most common causes of lost marks.

一定要先化简方程，合并同类项并把所有项移到一边。漏掉负号是失分最常见的原因之一。

Equation / 方程	a	b	c
3x² − 5x + 2 = 0	3	−5	2
x² + 4x = 0	1	4	0
2x² + 3 = 5x	2	−5	3

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3. Solving by Factorising | 因式分解法

usually the fastest method. For example, x² − 5x + 6 = 0 factorises as (x − 2)(x − 3) = 0.

当二次表达式能写成两个一次因式的乘积时，因式分解法通常是最快的方法。例如，x² − 5x + 6 = 0 可因式分解为 (x − 2)(x − 3) = 0。

zero, at least one factor must be zero. So x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

根据零乘积性质，如果两个因式的乘积为零，则至少一个因式为零。因此 x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。

(x − 2)(x − 3) = 0 → x = 2 or x = 3

y. For example, 2x² + 7x + 3 = 0 factorises as (2x + 1)(x + 3) = 0, so x = −1/2 or x = −3.

对于较难的情况，要仔细检查 a 和 c 的因数。例如，2x² + 7x + 3 = 0 可分解为 (2x + 1)(x + 3) = 0，所以 x = −1/2 或 x = −3。

or x = −3. Factorising only works neatly when the roots are rational or simple fractions.

平方差是重要的特殊情况：x² − 9 = 0 得到 (x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。因式分解只在根为有理数或简单分数时比较方便。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This method is very useful for finding the vertex and for solving equations when factorising is difficult.

配方法把二次表达式改写成 a(x + p)² + q 的形式。这种方法在寻找顶点以及因式分解困难时非常有用。

btract it. For example, x² + 6x + 5 = 0 becomes (x + 3)² − 9 + 5 = 0, so (x + 3)² − 4 = 0.

对于 x² + bx + c，取 b 的一半，平方，再加回并减去。例如，x² + 6x + 5 = 0 变为 (x + 3)² − 9 + 5 = 0，即 (x + 3)² − 4 = 0。

(x + 3)² = 4 → x + 3 = ±2 → x = −1 or x = −5

he whole equation by a. For example, 2x² + 8x + 3 = 0 can be written as 2(x + 2)² − 5 = 0.

如果 a ≠ 1，先把 a 从 x² 和 x 项中提取出来，或者整个方程除以 a。例如，2x² + 8x + 3 = 0 可写成 2(x + 2)² − 5 = 0。

tex is (h, k). Completing the square is therefore a bridge between algebra and graph work.

在顶点式 y = a(x − h)² + k 中，顶点是 (h, k)。因此配方法是代数与图像之间的桥梁。

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5. Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation in standard form. It is especially useful when factorising is not obvious. The formula is:

求根公式可以解任何标准形式的二次方程。当因式分解不明显时，它尤其有用。公式为：

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x = (−b ± √b² − 4ac) / (2a)

e a = 3, b = −5 and c = 2. The discriminant is b² − 4ac = (−5)² − 4 × 3 × 2 = 25 − 24 = 1.

例如，解 3x² − 5x + 2 = 0。这里 a = 3、b = −5、c = 2。判别式为 b² − 4ac = (−5)² − 4 × 3 × 2 = 25 − 24 = 1。

x = (5 ± √1) / 6 = (5 ± 1) / 6 → x = 1 or x = 2/3

Substitute carefully, especially when b is negative. Use brackets around negative values to avoid sign errors. If the discriminant is not a perfect square, leave the answer in exact surd form unless the question asks for decimals.

代入时要特别小心，尤其是 b 为负数时。负值要加括号以避免符号错误。如果判别式不是完全平方数，除非题目要求小数，否则应保留精确根式形式。

Use the quadratic formula when the equation does not factorise easily, when coefficients are large, or when the question asks for exact solutions or decimal answers to a given accuracy.

当方程不易因式分解、系数较大，或题目要求精确解或指定精度的小数答案时，使用求根公式。

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6. The Discriminant and Nature of Roots | 判别式与根的性质

ac. It tells you how many real roots the quadratic equation has, without solving it fully.

判别式是根号下的表达式：Δ = b² − 4ac。它可以在不完全求解的情况下告诉你二次方程有多少个实数根。

ated real root. If Δ < 0, there are no real roots, so the graph does not cross the x-axis.

如果 Δ > 0，有两个不同的实数根。如果 Δ = 0，有一个重根。如果 Δ < 0，没有实数根，因此图像不与 x 轴相交。

Discriminant / 判别式	Nature of roots / 根的性质	Graph / 图像
Δ > 0	Two distinct real roots / 两个不同实根	Crosses x-axis twice / 与 x 轴相交两次
Δ = 0	One repeated real root / 一个重根	Touches x-axis once / 与 x 轴相切
Δ < 0	No real roots / 无实数根	Does not cross x-axis / 不与 x 轴相交

= 0, so it has one repeated root. x² + x + 1 = 0 has Δ = −3 < 0, so it has no real roots.

例如，x² − 5x + 6 = 0 的 Δ = 1 > 0，所以有两个实根。x² − 4x + 4 = 0 的 Δ = 0，所以有一个重根。x² + x + 1 = 0 的 Δ = −3 < 0，所以没有实数根。

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

. If a > 0, the parabola opens upward. If a < 0, it opens downward. The larger the value of |a|, the narrower the parabola.

y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上。如果 a < 0，开口向下。|a| 越大，抛物线越窄。

= −b/(2a). The vertex lies on this axis and is the minimum or maximum point of the graph.

y 轴截距是 c，因为当 x = 0 时，y = c。对称轴是竖直线 x = −b/(2a)。顶点位于这条对称轴上，是图像的最低点或最高点。

Axis of symmetry: x = −b / (2a)

Substituting x = 2.5 gives y = 2.5² − 5 × 2.5 + 6 = −0.25, so the vertex is (2.5, −0.25).

对于 y = x² − 5x + 6，a = 1 > 0，所以抛物线开口向上。对称轴是 x = 5/2 = 2.5。代入 x = 2.5，得 y = 2.5² − 5 × 2.5 + 6 = −0.25，所以顶点是 (2.5, −0.25)。

ing the square: y = (x − 2.5)² − 0.25. This shows the minimum value is −0.25 when x = 2.5.

你也可以通过配方法求顶点：y = (x − 2.5)² − 0.25。这说明当 x = 2.5 时，最小值为 −0.25。

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8. Linking Roots, Vertex and Graph | 根、顶点与图像的联系

+ c. They are the values of x where the graph meets the x-axis, so y = 0 at those points.

ax² + bx + c = 0 的根是图像 y = ax² + bx + c 的 x 轴截距。它们是图像与 x 轴相交处的 x 值，因此在这些点 y = 0。

halfway between the roots: x = (p + q)/2. The vertex therefore lies on this vertical line.

如果根是 p 和 q，因式形式为 y = a(x − p)(x − q)。对称轴位于两根的中点：x = (p + q)/2。因此顶点就在这条竖直线上。

y = a(x − p)(x − q), axis: x = (p + q) / 2

lation. The graph crosses the x-axis at 2 and 3, and the vertex is below the x-axis at (2.5, −0.25).

对于 y = x² − 5x + 6，根是 x = 2 和 x = 3。对称轴是 x = (2 + 3)/2 = 2.5，与顶点计算一致。图像在 2 和 3 处穿过 x 轴，顶点位于 x 轴下方，即 (2.5, −0.25)。

The discriminant describes the same information graphically: two roots mean two x-intercepts, one repeated root means the vertex touches the x-axis, and no real roots means the parabola stays completely above or below the x-axis.

判别式在图像上表达同样的信息：两个根意味着两个 x 轴截距，一个重根意味着顶点与 x 轴相切，无实数根意味着抛物线完全在 x 轴上方或下方。

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9. Solving Quadratic Inequalities | 二次不等式

To solve a quadratic inequality, first solve the related quadratic equation, then

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