Algebra - Factorisation - Completing the Square.
Steps to follow.
1. Basic monic expression or equation.
Task: Complete the square on x2 - 6x + 5. | ||
Step 1: | Open a bracket and write an x. | (x |
Step 2: | Copy the sign before the x-term. | (x - |
Step 3: | Halve the number in front of the x (6 ÷ 2) = 3. | (x - 3 |
Step 4: | Complete the brackets and square. | (x - 3)2 |
Step 5: | Take off the number you just created - here 32 so subtract 9.
It is ALWAYS subtract!! |
(x - 3)2 - 9 |
Step 6: | Copy anything following the first 2 terms in the original - here +5. | (x - 3)2 - 9 + 5 |
Step 7: | Tidy up. | (x - 3)2 - 4 |
If you have an expression to factorise, you have completed your task. | ||
If you are solving the equation x2 - 6x + 5 = 0: | ||
Copy your factorisation and put it = 0. | (x - 3)2 - 4 = 0 | |
Solve in the normal way: | (x - 3)2 = 4
x - 3 = + 2 or - 2 ∴ x = 5 or x = 1 |
REMEMBER: look again at the videos if you are not sure of any of these steps.
2. Non-monic expression or equation.
Task: Complete the square on 2x2 - 7x + 3. | ||
Preliminary step: | Factorise out the coefficient of x2 | |
Step 1: | Inside the main brackets, open another bracket and write an x. | |
Step 2: | Copy the sign before the x-term. | |
Step 3: | Halve the number in front of the x (7/2 ÷ 2) = 7/4. | |
Step 4: | Complete the brackets and square. | |
Step 5: | Take off the number you just created - here (7/4)2 so subtract (49/16). Its ALWAYS subtract!! |
|
Step 6: | Copy anything following the first 2 terms in the original - here +5. | |
Step 7: | Tidy up. | |
If you have an expression to factorise, you have completed your task. | ||
If you are solving the equation 2x2 - 7x + 3 = 0: | ||
Copy your factorisation and put it = 0
and then solve in the normal way: |