Completing the square is a useful method for solving quadratics that cannot be factorised and has other applications such as being a way of simplifying more complex expressions.
First let’s take a generic quadratic equation:
Where b and c are both constants. To complete the square, we need to turn the quadratic into the following form:
Where d can be found by dividing the coefficient of x (b) by 2:
and e can be found by subtracting d2 from c:
This can be summarised as:
For example, if we look at the quadratic x2 + 4x + 5 we know that b = 4 and c = 5.
Therefore, we can work out d and e as follows:
To get our completed square as:
Or we can put it into our summarised formula to get the exact same answer:
When the coefficient of x2 is Greater than 1
Quite often the coefficient of x2 is not equal to 1:
Where a, b and c are all constants. However, we can still complete the square of these quadratics. All you need to do is factorise the quadratic by dividing through by a:
We can now complete the square for everything inside the bracket as above, d and e however, become:
This can be summarised as:
We can simplify this too:
For example, if we look at the quadratic 2x2 + 8x + 3 we know that a = 2, b = 8 and c = 3. Inputting these values into the formula we get:
Worked Examples
Example 1
Complete the square for the following quadratic:
x2 + 4x – 7
Example 2
Complete the square for the following quadratic and solve for x:
x2 – 4x – 2 = 0
Example 3
Solve 3x2 – 6x + 2 = 0 by completing the square.
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