1. Completing the square
Tuesday, 23 April 2019

2. Minimum value here would be –14
Revision of completing the square Example Complete the square for
Minimum value here would be –14

3. Complete the square for the following (i) (ii) (iii)
Example
Complete the square for the following
(i)
(ii)
(iii)
(i)
(i)
(i)
This one shows that the quadratic is always positive. i.e. Always 10 or more.

4. Example
Express in the form

5. Example
Express in the form

6. From this the minimum value is 3
A Level Question
Express in the form , where a, b and c are constants whose values are to be found [3]
Use your answer to part (a) to find the greatest value of [2]
(a)
From this the minimum value is 3
(b)
Therefore maximum value =

7. Here the minimum value is 5 when x = 2
A Level Question
Express in the form , where the values of the constants a, b and c are to be found.
Hence, sketch the graph of , indicating the coordinates of its stationary point. [5]
Here the minimum value is 5 when x = 2 x y
(2, 5)

8. The Circle Centre the origin

9. The Circle Centre (0, 0) radius r

10. Angles between a tangent and its radius is always 90º
To find the gradient of any tangent at a specific point on a circle
find the gradient of the radius, then
the gradient of the tangent using knowledge of perpendicular lines

11. Find the gradient of the tangent to the circle at the point
Example
Find the gradient of the tangent to the circle at the point
Gradient of the radius =
Gradient of the tangent =
(2, –3)

12. Completing the square and Circles Introduction
In questions 1 – 5 complete the square 1. 2. 3. 4. 5.

13. 9. Find the gradient of the tangent to the circle at the point
In questions 6 – 8 express in the form , where a, b and c are constants and state the minimum value. 6. 7. 8.
9. Find the gradient of the tangent to the circle at the point
10. Find the gradient of the tangent to the circle at the point
Minimum =
Minimum =
Minimum =
Gradient =
Gradient =