Does a Quadratic Equation know what context it is in?

Often when students are solving quadratic questions in A-level maths they employ different techniques:

• Quadratic formula

• Completing the square

• Factorising

However quadratics can appear in many A-level topics such as:

• Discriminant problems

• Trig solving problems

• Sequences and Series

• Logs

• Exponentials

Here are some examples of hidden quadratics:

In both of these examples you will need to find a replacement for tan(x) and for the 2 to the power of x. Once found you can rewrite the whole expression as a quadratic and solve to find a value for tan(x) and 2 to the power of x.

This is not really the point of the post however. Most learners are able to recognise and solve quadratic trig questions (and others) but often lose marks because they fail to see the context in which the solutions of the quadratic equation sits.

In other words- the quadratic equation does not know the context of the question, YOU HAVE TO APPLY THE CONTEXT TO PICK THE CORRECT SOLUTION!

Here is an example of an incorrect solution to a quadratic in context question:

This particular topic is related to recurrence relations in Year 13

The student had achieved 3/4 marks for correctly getting the first 3 terms, adding them to make 12 and rearranging to solve a quadratic in k.

Why did they lose one mark?

The student did not TEST their k values in the context of the question! Have a look at the mark scheme below:

Model Mark scheme for this question

You can see that the student should have tested each k value in the context of the question. In other words, do either one fail to uphold the fact that each term is different?

You can clearly see K= -2 is the correct answer after doing this!

Here is a question which further highlights the point :

Trig quadratic equation

Here is the mark scheme, you can clearly see that one of the values is not valid as the sin(x) graph does not have solutions for beyond 1 or -1.

Conclusion

When solving quadratic expressions within context (trig, seq and series, etc) then don't just state all the solutions. Do check each one for validity in the context of the question, functions being used or just by looking at the range of valid values accepted.