**Question**

**Introduction**

In this
secondary 2 (approx. grade 8~9) algebra question, we are essentially asked to
make

Note that

So observe that in the given equation, the RHS is non-negative. Hence the LHS which is just

*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**

*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 3*a*–*k*^{2}__>__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.**Solution**

**Remarks**

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 “

*x*^{2}= 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*x*^{2}= 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)

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