Quadratic and fractional equations

In the second year we raise the level: equations are no longer only linear, but can have the unknown squared. This changes everything, because the solutions can be two, just one, or none in the real field. The chapter revisits fractional equations, introduces the canonical form $ax^2+bx+c=0$ and the resolving formula with the discriminant $\Delta$, presents Vieta’s formulae relating the sum and product of the roots to the coefficients and their application to symmetric systems. In the final part it goes beyond the minimum syllabus: Cardano’s formula for cubics, continued fractions, infinite nested radicals and geometric series of areas, topics in which quadratic algebra becomes a surprisingly powerful tool.

Sections

• Revision: fractional equations
• Quadratic equations
• Vieta’s formulae
• Symmetric systems
• Cardano’s formulae for cubic equations
• Continued fractions
• Infinite nested radicals
• Geometric series of areas

Further reading

• Tartaglia, Cardano and the cube formula
• François Viète, the inventor of symbols

Exercises

• All the worked exercises of the chapter