4/17/2024 0 Comments Solve complex quadratic equationThe third type of disguised quadratic equation is found in trigonometric equations, such as 5 cos 2 θ cosec θ = 2. These questions will often explicitly ask for the solution to be given as an exact value in a specific form such as ‘giving your answer in the form x = a log b c, where c > 1’, with the final accuracy mark awarded for manipulating into this required form. While factorising 3e 2 x + 14e x – 5 = 0 to (3e x – 1) (e x + 5) is a valid method, examiners report that a substitution such as X = e x often yields fewer mistakes.Įquations such as 3 2 x – 3 x – 2 = 0 can be solved by different routes, but again success is seen more often when using a substitution such as X = 3 x. Students should be able to recognise the standard three term quadratic format when the equation has the unknown as the exponent. Note that it is good practice to make a clear distinction between the initial equation variable and the variable used in the substitution version. Graphing can help demonstrate when roots may need to be rejected (see disguised quadratic on Desmos). Here it is important to not only state ‘ X ≠ -3.5 since x 2 ≥ 0’, but also to remember to finish the solution with ‘ x 2 = 1, so x = ± 1’. (2√ y + 7) (√ y – 1) = 0.Ī similar approach can be taken for the equation 2 x 4 + 5 x 2 – 7 = 0, using a substitution such as ‘Let X = x 2’. In this case the solution would need to include the statement ‘ x ≠ -3.5 since √ y ≥ 0’.Įxaminers have noted that students make fewer arithmetic errors when using the clear substitution method rather than trying to find the factorised form, i.e. The equation 2 y + 5√ y – 7 = 0 can be reduced to the standard quadratic by using the substitution ‘Let x = √ y’.Ĭandidates need to be aware that explicitly stating the substitution or the justification for rejecting roots are part of a clear mathematical argument. In mark schemes, ‘www’ (‘without wrong working’) means that subsequent accuracy marks cannot be awarded when the method is incorrect, resulting in M0A0. Similarly, attempts to ‘reverse engineer’ the completed square format from the calculator often result in sign errors with the term either inside or outside the bracketed term, or missing the leading multiplying factor.Įxaminers will check the working that is presented alongside a final answer. Here, misconceptions about the link between factors and roots are often seen, such as either the incomplete ( x – 1) ( x + 3.5) = 0 or incorrect signs ( x + 1) (2 x – 7) = 0. Some candidates appear to ‘reverse engineer’ their factorisation from the calculator. Over the years, examiners have reported students making more errors when attempting to substitute values into the quadratic formula than with the other two methods. Students will be familiar from GCSE (9-1) Maths with equations such as 2 x 2 + 5 x – 7 = 0, a standard three term quadratic that could be solved by: There is no restriction in using this function, although students may sometimes be able to solve standard quadratic equations by inspection just as quickly. Many of the calculators allowed for maths exams include a function to solve quadratics. In A Level Maths there is an emphasis on mathematical reasoning how the answer is obtained is just as important as the answer itself. Using the same set of techniques developed for quadratic equations we can solve a range of equations, such as polynomial, exponential and trigonometric equations. I’m now going to look at some of the issues seen by examiners as the complexity of the quadratic increases. In my previous blog I looked at using graphs to investigate quadratic functions. A Level Maths: Solving complex quadratic equations
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