This course is about equations in which the unknown appears squared. It begins with the standard form and what a root is, then teaches three ways to solve: factorisation by splitting the middle term, completing the square, and the quadratic formula derived from it. The discriminant shows whether the roots are real, equal or irrational without solving. The relations for the sum and product of the roots lead to forming equations from their roots and to evaluating expressions such as the sum of the squares of the roots. The course ends with equations that reduce to quadratics and with word problems on area, speed, work, age and cost.
Middle Years
Quadratic Equations and Roots
Quadratic equations and their roots: factorisation, completing the square, the quadratic formula, the discriminant, sum and product of roots, equations that reduce to quadratics, and word problems.
- 9 lessons
Overview
Lessons
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Quadratic equations and their roots
The standard form ax² + bx + c = 0, bringing equations to standard form, what a root is, checking roots, and writing word statements as quadratic equations.
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Solving by factorisation
The zero product rule, splitting the middle term, equations with no constant or no middle term, surd coefficients, and equations that need tidying first.
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Completing the square
Adding the square of half the coefficient of x to solve any quadratic, recognising when there are no real roots, and finding least and greatest values.
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The quadratic formula
Deriving the formula x = (−b ± √(b² − 4ac)) / 2a by completing the square, using it carefully with signs and surds, and choosing the best method.
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The discriminant and the nature of roots
How b² − 4ac decides whether the roots are distinct, equal, rational, irrational or not real, and how to find coefficients that give equal or real roots.
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Sum and product of roots
The relations α + β = −b/a and αβ = c/a, forming an equation from its roots, finding one root from the other, and roots in a given ratio or difference.
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Symmetric expressions in the roots
Finding α² + β², α³ + β³, 1/α + 1/β and similar expressions from the sum and product, and forming equations whose roots are related to the given roots.
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Equations reducible to quadratic form
Equations in x⁴ and x², repeated expressions, fractions, square roots and unknown powers, solved by a substitution and checked in the original equation.
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Word problems on quadratic equations
Four steps for word problems, with problems on numbers, area, right triangles, speed and time, work, ages and cost, keeping only the roots that make sense.