SM1.4 Quadratic equations
You can solve a quadratic equation by factorising and by the quadratic formula, use the discriminant to say how many real roots it has, and solve word problems that lead to quadratics.
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A quadratic equation asks where a curve meets a line, and it can have two answers, one, or none. Factorising finds the answers when they are whole and friendly; the quadratic formula finds them always; and the discriminant, the part under the square root, tells you how many to expect before you do any work. The word problems here are the first ones where an equation can return an answer the story must refuse, such as a negative length, and noticing that is part of the answer.
Work through these
Solve a set of quadratics by factorising, keeping to the ones where factors come easily
Factorising is the fast route and the one that builds number sense, but it only works when the roots are friendly. Learning to recognise when it will work, and to stop forcing it when it will not, is itself the skill.
Solve the awkward ones with the quadratic formula, writing the discriminant separately first
Writing the discriminant on its own line before the rest of the formula means you learn its value while it still means something: how many roots are coming. It also isolates the place where most arithmetic slips happen.
For several equations, state the nature of the roots from the discriminant without solving
Positive, zero or negative under the root is a three-way verdict that examinations ask directly and repeatedly. Answering it without solving proves you know what the discriminant is, rather than where it sits in the formula.
Solve two word problems that lead to quadratics, and reject any root the story cannot accept
A quadratic often returns two answers and the situation usually allows one. Saying explicitly why the other is rejected, an age cannot be negative, a side cannot be zero, is what a full answer looks like.
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