The remainder of this topic is higher only. Before reading on, make sure to click here (https://mathsmadeeasy.co.uk/gcse-maths-revision/completing-square-gcse-maths-revision-worksheets/) to ensure you understand how to complete the square of a quadratic.
Before we get right into an example, we must make the following observation. If we’re looking for a minimum point on a graph (remember, the turning point of a positive quadratic is the minimum point), then we’re looking for the smallest value of y that the graph reaches. Since the graph is determined by its equation, this is also the smallest value of y that the equation can take. Let’s see a couple of examples.
Example: Use completing the square to find the coordinates of the turning point of y=x^2+4x-12.
Firstly, we must complete the square. Given that half of 4 is 2, our quadratic becomes
Now, this is a positive quadratic, so we are looking for a minimum point. To find the coordinates of the minimum point, we will ask: what is the smallest value this quadratic can take?
The right-hand side of our equation comes in two parts. The first part, (x+2)^2, is always going to be positive no matter what x we put in, because the result of squaring a real number is always positive. If (x+2)^2 is never negative, then the smallest number it can be is zero, meaning that the minimum value of the quadratic is y=0-16=-16.
Now we know the y coordinate of our minimum point, what is the x coordinate? The question is, what value of x must we put into our equation to make (x+2)^2=0? The answer is x=-2. Therefore, the coordinates of our minimum point are (-2,-16). Try plotting the graph to see for yourself.
Example: Complete the square to find the coordinates of the turning point of y=2x^2-20x+14.
To complete the square on this, we first take a factor of 2 out of the whole quadratic:
Now, we complete the square on the inside section. Half of 10 is 5, so we get
Having the extra number in front of the bracket doesn’t actually change anything. We’re still looking for a minimum point, and the minimum value that 2(x+5)^2 can take is zero, so the minimum value of the quadratic is y=0-36=-36.
The x coordinate that is required to get this minimum value has to make the expression 2(x+5)^2 equal to zero, so it must be x=-5. Therefore, the coordinates of the turning point (minimum point) are (-5, -36).
If you aren’t sure about this, have another read of these two examples to make sense of it. Once you’ve got used to it, it just becomes a process of reading off the coordinates of the turning point once you have completed the square.
So, the main thing to remember is, when the result of completing the square is
the turning point of y=a(x+b)^2+c has coordinates (-b, c).