Quadratics and inequalities
도입됨| English | 의미 |
|---|---|
| discriminant/dɪˈskrɪmɪnənt/ | 판별식 |
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.
The discriminant of x²-6x+9=0 is 0. What does that tell you?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
b²-4ac=36-36=0 means the parabola just touches the axis: one repeated root, here x=3. No real roots needs a negative discriminant.
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 condition is x(10-x)≥21, so x²-10x+21≤0. Factor (x-3)(x-7)≤0. The upward parabola is nonpositive between its roots, giving 3≤x≤7. The maximum area is 25 at x=5.
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The discriminant is (-6)²-4×1×9=0.
A positive discriminant means that a quadratic has no real roots.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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
- Use the vertex to interpret an optimum. Check an endpoint and a point between the roots; the algebra and graph should tell the same story.
- 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).
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Complete the square: x(10-x)=25-(x-5)². The maximum is 25.
Match each discriminant value to the roots of the quadratic.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The discriminant counts real roots before any solving; it tells you what kind of answer to expect.
Use this in your course
- edexcel IAL mathematics; official unit P2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- 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.