**'I think of a number, multiply it by itself then add 9 and the answer is 90. What number was I thinking of?'**

You may already be familiar with this type of puzzle and how it can be expressed **algebraically**.

These sorts of problems are best solved by writing them as **equations**.

If we use **x** (or any letter) to stand for original number, we can write the puzzle without using any words but using numbers, letters and symbols as: **x² + 9 = 90 **

Remember, that when we multiply a number by itself, we are **squaring **it.

Any equation which contains an **x²** as the** highest power of x **(so we have no x³ for example) is called a **quadratic equation**.

Solving this special type of equation is a very important area of maths.

Now, to find out what **x **is, we need to **solve** the equation.

This means getting it into the form: **x = ...**

**Remember that whatever we do to one side of the equation, we must do to the other. **

So, firstly we need to remove the** + 9** by doing the inverse to both sides which is **- 9**.

This leaves us with:

**x² = 81**

Now, we need to remove the **'squaring'** by doing the inverse of this 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 **both** **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 can set out our working like this:

Let's try another example now.

Here is another equation: **2x² + 2 = 100**

In order to solve this equation, first, we must **subtract 2** from both sides:

**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 **× 2** next which is **÷ 2**:

**x² = 49**

Finally, we **square root **both sides to give:

**x = ±7**

We set out our working like this:

If our last step here does not have a whole number square root, we will need to use a calculator.

There are some questions like this in the activity, so you should grab a calculator before you start.

In this activity, we will solve quadratic equations with three terms by rearranging and calculating square roots of whole numbers and decimals.