Increase your marks – avoid these common errors…

Let’s face it: there will almost certainly be questions, or parts of questions, in your exam papers that you just can’t do. Maybe you’ve forgotten the method, or the algebra is too hard, or you don’t understand what is being asked, or you simply get stuck. OK, so you probably won’t get 100%. But there are also lots of very common slips which lead to unnecessary marks being lost; do your very best not to fall into these numeric and algebraic traps.

1.  Every number has two square roots.
Solve this equation:          x^2 = 16
If your answer is x=4 then that is only partly right. -4 is also the square root of 16, so the correct solution is x = \pm 4

Do this every time you square root a number, but do check whether a question limits solutions to positive numbers only.

2.  When substituting numbers for letters, put negative numbers in brackets.
For example, work out x^2-5 if x=-3. If we write -3– 5, then the answer would be -9 – 5 = -14, and that is what your calculator would give as well. Why is it wrong? Because when we substitute -3 for x we need to find the square of -3, which is 9. We should have used a bracket: (-3)– 5 = 9 – 5 = 4.

Always use brackets for negative numbers: in particular, this will help you avoid mistakes when working out binomial expansions.

3.  Numerators and denominators of fractions – think “brackets”.
Look at the following equation and its (incorrect) solution:

\begin{array}{l} \frac{{4x}}{{x + 3}} = 2\\ 4x = 2x + 3\\ 2x = 3\\ x = 1.5 \end{array}

The working goes wrong on the second line because the 2 should be multiplying the x + 3, not just the x. The confusion arises because the fraction bar does away with the need to use brackets in numerator and denominator. If I put the brackets back in, we get:

\begin{array}{l} \frac{{4x}}{{(x + 3)}} = 2\\ 4x = 2(x + 3)\\ 4x=2x+6\\ 2x = 6\\ x = 3 \end{array}

Whenever you have an expression with more than one term anywhere in a fraction, put brackets around it for safety.

4.  Think of brackets as “locked doors.”
What do I mean by this? Simply, there is no communication possible between things outside brackets and things inside brackets. To get at the contents of brackets you need to multiply out, or use other rules as appropriate. Examples (and, as in point 3 above, I have inserted brackets into fractions):

  • \frac{{(x+3)}}{{2}}=4   …  you cannot subtract 3 from both sides of the equation.
  • \frac{{(x-6)}}{{x}}   …   you cannot cancel the x‘s (this algebraic fraction cannot be simplified).
  • sin(2x+30)  …   this is not the same as sin2x+sin30. Use the rule for expanding sin(A+B).
  • (x+5)^2  …  does not equal x^2+25. Multiply out (x+5)(x+5).

Always think about how to deal with the contents of brackets.

5.   Negative signs outside brackets.
So far I’ve dealt with negative signs and with brackets. Combining the two, don’t forget how to deal with a negative sign in front of a bracket, or a negative number which multiplies a bracket. Everything in the bracket is affected:

  • 12-3(x+2) = 12-3x-6=6-3x
  • -(x-2)^2=-(x^2-4x+4)=-x^2+4x-4

Negative sign, or negative number, in front of a bracket: beware!

6.   “Two negatives make a positive” – do they?
Yes, if you’re talking multiplication or division:

  • (-2) \times (-3)=6
  • (-5) \div (-2)=2.5 or \frac{{-5}}{{-2}}=2.5
  • If a = -6x2 and b = -3x , work out \frac{{a}}{{b}}.       Answer: 2x.

So, when multiplying or dividing, you know in advance what the sign of the answer will be. Simply put, if both signs are the same, the answer will be positive; if the signs are opposite, the answer will be negative.

But, when adding or subtracting, the sign of the answer will depend on the numbers you start with. The only rules to remember are that:

  • Adding a negative number is the same as subtraction.
    eg:  6 + (-4) = 6 – 4 = 2
  • Subtracting a negative number is the same as addition.
    eg:   8 – (-3) = 11

Now, when doing the sum, think of a number line. Your start point is the first number; then for addition go right, for subtraction go left.

Examples:

  • (-5) + 8.     Start at -5, go right 8. The answer is 3.
  • (-11) + 2.    Start at -11, go right 2. The answer is -9.
  • 4 + (-7).      Start at 4, go left 7. The answer is -3.
  • -6 – (-2).     Start at -6, go right 2. The answer is -4.
  • -5 – 3.          Start at -5, go left 3. The answer is -8. (This is where people often think: “Two negatives make a positive, so the answer must be +8, or +2”).

The rules for addition and subtraction are completely different to the rules for multiplication and division.

If you’ve got more ideas about common slips that you want to share, do add a comment to this post. I’ll collect them together and write a part 2 early next year.

 

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