Quadratics and inequalities · Quadratiques et inégalités
| English | Français |
|---|---|
| discriminant/dɪˈskrɪmɪnənt/ | discriminant |
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? · Quelle description définit correctement le discriminant ?
The quantity b²-4ac that determines the real roots of ax²+bx+c=0. · La quantité b²-4ac qui détermine les racines réelles de 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 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 · Quadratiques et inégalités
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. · La courbe redimensionne l'axe vertical d'un facteur positif 1/4. Vérifier pourquoi ses racines restent 3 et 7.
Find the smaller root of x²-10x+21=0. · Trouve la plus petite racine de x²-10x+21=0.
Factorise to (x-3)(x-7)=0. The smaller root is 3. · Factorise en (x-3)(x-7)=0. La plus petite racine est 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. · Trouve le discriminant de x²-6x+9=0.
The discriminant is (-6)²-4×1×9=0. · Le discriminant est (-6)²-4×1×9=0.
A positive discriminant means that a quadratic has no real roots. · Un discriminant positif signifie qu'un polynôme du second degré n'a pas de racines réelles.
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. · Multiplier une inégalité par un nombre négatif inverse son sens. Un croquis doit indiquer quel côté de chaque racine satisfait l'inégalité. La géométrie peut restreindre davantage x.
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). · Trouver la valeur maximale de x(10-x).
Complete the square: x(10-x)=25-(x-5)². The maximum is 25. · Compléter le carré : x(10-x)=25-(x-5)². Le maximum est 25.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- 7357 · A-level · B. 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.