'I think of a number, multiply it by itself then add 9 and the answer is 90. What number was I thinking of?'
If you have completed any of the Level 1 activities on linear equations you will be familiar with this type of puzzle. But if you haven't, do not worry, all will be explained. These sorts of problems are best solved by writing them as equations. If I use x (or any letter) to stand for the number I thought of, I can write the puzzle without using any words but using numbers, letters and symbols as:
x² + 9 = 90
Remember, when we multiply a number by itself we are squaring. Any equation which contains an x² as the highest power of x (no x³ for example) is called a quadratic equation, and solving them is a very important area of maths. Now, to find out what number x is I solve the equation. That means getting it in the form x = ......, by doing the same to both sides. So, firstly we need to remove the +9 by doing the inverse to both sides which is -9. This gives us
x² = 81
Now, I need to remove the 'squaring' by doing the inverse to both sides which is 'square rooting'. This gives us
x = 9
Now strictly speaking there are two square roots of 81. This is because when you square a negative number you get a positive, so 9² = 81 and (-9)² = 81. So the square root of 81 could be 9 or -9. Usually, we are just interested in the positive square root, but it is always safest to include both. Remember, the symbol for square root is √ so we write
√81 = 9 or -9 or √81 = ±9 (the ± symbol means the answer can be positive or negative)
We set the working out to this as follows:
Here is another equation:
2x² + 2 = 100
In order to solve this equation, first, we must subtract 2 from both sides. This will give us 2x² = 98. Now we need to be careful about the order we do the next two inverses. 2x² means 2 times x² so we need to apply the inverse of x2 next which is ÷2. This gives us x² = 49. Finally, we square root both sides to give x = ±7. We set out the working as follows.
If the last step does not have a whole number square root we will need to use a calculator. There are some questions like this in the exercise. Ready to try some now? Here we go!