Factorising and solving simple quadratics · Foundation
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| discriminant/dɪˈskrɪmɪnənt/ | 判别式 | pàn bié shì |
Which widths make enough space?
- A rectangular enclosure has area x(10-x). What widths give at least 21 square metres?
- This lesson studies discriminant 判别式: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
哪些宽度能留出足够空间？
- 一个矩形围栏的面积为x(10-x)。求能提供至少21平方米面积的宽度。
- 本课学习判别式（discriminant）：表达式b²-4ac，用于确定方程ax²+bx+c=0的实数根。
Choose the mathematical structure
- Expand brackets and factorise simple quadratics. Solve by setting each factor equal to zero, and use a graph to interpret the roots.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
选择数学结构
- 展开括号并对简单二次三项式进行因式分解。令每个因式等于零来求解，并利用图像解释根的含义。
- 计算前请先明确允许的输入项和单位。方程应表达关系本身，而不仅仅是记录计算器按键过程。
Which description correctly defines discriminant?
The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
x²-10x+21=(x-3)(x-7). Hence the equation x²-10x+21=0 has roots 3 and 7. Check each root by substitution and mark both intercepts on the graph.
通过验证案例进行推导
- 将结果与初始量进行比对。代入原始关系式，或在适当情况下对比图表与数值答案。
x²-10x+21=(x-3)(x-7)。因此，方程 x²-10x+21=0 的根为 3 和 7。通过代入法逐一验证每个根，并在图像上标出两个截距点。
Factorising and solving simple quadratics
y=(x²-10x+21)/4
The graph rescales the vertical axis by a positive factor 1/4. Check why its roots remain 3 and 7.
Find the smaller root of x²-10x+21=0.
Factorise to (x-3)(x-7)=0. The smaller root is 3.
Test a tempting shortcut
- Multiplying an inequality by a negative number reverses its direction. A sketch must show which side of each root satisfies the inequality. Geometry may restrict x further.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A positive discriminant means that a quadratic has no real roots. This claim is false. Explain which definition or assumption it violates.
检验一个诱人的捷径
- 不等式两边同乘负数时，不等号方向必须改变。示意图须标出满足不等式的区域位于每个根的哪一侧。几何约束可能会进一步限制x的取值范围。
- 当捷径失效时，找出其违背的假设。保留精确值直到题目要求的最终舍入步骤。
判别式为正值意味着二次函数没有实数根。此说法错误。请说明它违反了哪一定义或假设。
Find the larger root of x²-10x+21=0.
The factorised roots are 3 and 7; the larger is 7.
A positive discriminant means that a quadratic has no real roots.
Multiplying an inequality by a negative number reverses its direction. A sketch must show which side of each root satisfies the inequality. Geometry may restrict x further.
Interpret a new situation
- This Foundation/Core lesson uses factorisation and graphical roots; the discriminant, quadratic formula and quadratic inequalities are reserved for the advanced tier.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
解读新情境
- 本基础/核心课程使用因式分解与图像求根；判别式、求根公式及二次不等式保留给进阶级别。
- 完整解答需给出数学结果并解释其含义。检查其在所述背景下是否可行。
Find (5-3)(5-7).
Substitute 5: (5-3)(5-7)=2×(-2)=-4.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 4MA1 · Foundation · 2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The quantity b²-4ac that determines the real roots of ax²+bx+c=0. Choose the relationship, show the method, check its assumptions and interpret the result.
将其应用于你的课程
- 4MA1 · 基础级 · 2. 在分配拓展内容前，请匹配目标层级和教学大纲要求。
- 在给出最终答案前先展示解题方法，并遵循试卷的计算器及公式使用规则。通过定位第一个无效步骤来复盘错误答案。
表达式b²-4ac，用于确定方程ax²+bx+c=0的实数根。请选择正确的关系，展示推导方法，检查其适用条件并解释结果。