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THE VISUAL LEARNING STUDIO
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35 visual explanations across 29 chapters, with topic maps throughout the library.
The chapter index will grow as additional notes are prepared. Chapters shown here are a learning collection, not a claim that the entire exam syllabus is complete.
SEE IT. CONNECT IT. EXPLAIN IT.
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Algebraic Expressions and Quadratic Equations — lesson overview
Expressions, identities and equations
An expression represents a mathematical quantity. An equation asserts equality; a solution is a value that makes that equality true.
An identity holds for every permissible value of its variables, such as (a+b)² = a² + 2ab + b². An equation such as x² = 4 is true only for specific x values.
The domain matters. Dividing by an expression assumes it is nonzero; ignoring that condition can remove valid solutions or introduce invalid reasoning.
Quadratic structure
A quadratic equation in one variable can be written ax² + bx + c = 0 with a ≠ 0. Its degree is two.
The quadratic formula is x = (−b ± √(b² − 4ac))/(2a). The discriminant D = b² − 4ac determines the nature of real roots.
D > 0 gives two distinct real roots; D = 0 gives a repeated real root; D < 0 gives no real roots, though complex roots exist.
Worked example
Solve x² − 5x + 6 = 0. Factorising gives (x−2)(x−3) = 0, so x = 2 or x = 3.
The zero-product rule applies because the product equals zero. From (x−2)(x−3) = 4 one cannot simply set each factor equal to zero.
For ax² + bx + c = 0, the sum of roots is −b/a and their product is c/a, counting roots appropriately. Here the sum is 5 and product 6.
