Quadratics and inequalities · Higher
| 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.
Choose the mathematical structure
- Factor where possible; otherwise complete the square or use the quadratic formula. A quadratic inequality needs the sign on intervals, not only the roots. The discriminant identifies repeated or missing real roots.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
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.
The equation x²-10x+21=0 factorises as (x-3)(x-7)=0, giving roots 3 and 7. Complete the square: x²-10x+21=(x-5)²-4, giving turning point (5,-4) and symmetry line x=5. The formula x=[-b±√(b²-4ac)]/(2a) also gives (10±4)/2=3,7. For 2x²+5x-3=0, the discriminant is 49 and roots are (-5±7)/4=1/2,-3. A graph gives approximate roots when exact factorisation is inconvenient. For an enclosure with area A=x(10-x), complete the square to get A=25-(x-5)². The greatest area is 25 at x=5, within 0<x<10.
Quadratics and inequalities
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.
Find the discriminant of x²-6x+9=0.
The discriminant is (-6)²-4×1×9=0.
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
- AQA A11/A18 Higher interprets roots, intercepts and turning points, completes the square and uses the quadratic formula, including equations needing rearrangement. Factorisation and substitution check each result.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the maximum value of x(10-x).
Complete the square: x(10-x)=25-(x-5)². The maximum is 25.
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
- 8300 · Higher · 3.2. 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.