Factorising and solving simple quadratics · Core · Factorisation et résolution d'équations quadratiques simples · Fondamental
| 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
- 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.
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.
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.
Factorising and solving simple quadratics · Factorisation et résolution d'équations quadratiques simples
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 larger root of x²-10x+21=0. · Trouver la plus grande racine de x²-10x+21=0.
The factorised roots are 3 and 7; the larger is 7. · Les racines factorisées sont 3 et 7 ; la plus grande est 7.
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
- 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.
Find (5-3)(5-7). · Calculer (5-3)(5-7).
Substitute 5: (5-3)(5-7)=2×(-2)=-4. · Substituer 5 : (5-3)(5-7)=2×(-2)=-4.
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
- 9260 · Core · 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.