Friday, November 13, 2015

[S2_20151112CSSR] Change of Subject and Square Roots


     In this secondary 2 (approx. grade 8~9) algebra question, we are essentially asked to make  k  the subject.  This means that through a series of algebraic manipulation, we arrive at a final equation in which  k  appears alone on one side (conventionally the LHS) and all the other “stuff” on the other side.  This question looks challenging firstly because k  appears in more than one place and then also we need to deal with the square root.

The Square Root
     Note that the principal square root (or simply “the square root”) is by the modern definition non-negative i.e. zero or positive.  Of course, what goes under the square root must also be non-negative, otherwise it would not even make sense as a real number. 
     So observe that in the given equation, the RHS is non-negative.  Hence the LHS which is just  k,  must be non-negative.  It is tacitly understood that  3ak2 > 0  for the square root to make sense.  
     To get rid of the square root, we can square both sides of the equation.  After that we bring all the terms with  k  to the LHS.  Finally, we need to “unsquare” both sides by taking square roots.  The solution takes only about 5 steps, as shown below.


     There is no need for  ±  in the final line because we already know that  k  is non-negative (k  is zero or positive).  Here is something that students and even teachers / tutors can get confused over.  Modern mathematics tends to take a “function” approach in which each expression can take only a single unambiguous value.  “ Ö ”  may be regarded as a function with the non-negative reals as domain and the non-negative reals as range.  Although in traditional parlance, we say things like the “square roots” of  9 are 3 and -3, once you see the  “ Ö ”  symbol (apart from the “±”), it is the result of a calculation and the result is by definition non-negative.
     Another thing that people get confused over is: What about the ± symbol ?  Note that  ±  by itself is actually meaningless!  Something like  ±3  is just a short-cut for lazy people to say “the answer is 3 or -3”  (and we tend to be lazy, don’t we?).  But this is the result of solving an equation like “x2 = 9” when  x  is a real number with no other restrictions.  This equation has two roots: 3 and -3.  If  x  is known to be non-negative, then  x = 3  is the only solution.  Of course solving an equation involves calculation.
     So what is the difference between solving and mere calculation?  Solving is a process of finding values for unknowns and it usually involves a more than one step and it may include calculation.  When you see something like  Ö9   you are just calculating, and there is only one answer.  But when you see something like “Find the values of  x  such that  x2 = 9” you are solving.  There is an unknown (e.g.  x)  and you are supposed to find number(s) that you can plug into  x  to satisfy that equation.  After you calculate  Ö9 = 3, you still need to write “x = 3 or x = -3” or its short form “x = ±3”.  I hope this clears the confusion.
     Remember “Ö something non-negative” Þ one non-negative answer.  “find / solve something” Þ maybe more than one answer.

H04. Look for pattern(s)
H05. Work backwards
H10. Simplify the problem
H11. Solve part of the problem
H13* Use Equation / write a Mathematical Sentence

Suitable Levels
Lower Secondary Mathematics (Secondary 2 » Grade 8/9)
GCE ‘O’ Level “Elementary” Mathematics (algebra, revision)
* other syllabuses that involve algebra

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