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Everything You Need To Know About Completing The Square!

07.01.2023

Question: Express xˆ2-8x+5 in the form (x-a)ˆ2-b


To express the quadratic expression xˆ2-8x+5 in the form (x-a)ˆ2-b, we can use the method of Completing The Square. 


Completing The Square is a mathematical technique used to solve quadratic equations and find the inverse function when we cannot rearrange for the unknown. When Completing The Square we rewrite the quadratic with the unknown in one place! This allows us to solve any quadratic without needing to factorise. Completing The Square also allows us to sketch a quadratic. However, we will cover that in another blog post.

Completing The Square is a process applied to the xˆ2 and x term - the constant at the end just follows along on each line. Generally speaking, a quadratic of the form xˆ2+2ax is equal to (x+a)ˆ2- aˆ2 (just expand it and you will see that they are the same). To Complete The Square we introduce a bracket squared and write x, and then half the coefficient of the x term of our original quadratic. We then subtract that halved coefficient squared. Therefore, if we have a quadratic in the form xˆ2+2ax+b we have the completed square of (x+a)ˆ2-aˆ2+b. Notice how the +b just follows along. 


So to write xˆ2-8x+5 in completed square form, we take half of the coefficient of x. In this case 8÷2=4.

We then insert it into our squared brackets and then subtract this halved coefficient squared. Don't forget the +b that carries at the end! 

(x-4)ˆ2-4ˆ2+5

Now simplify! We are left with: 

(x-4)^2-16+5 = (x-4)ˆ2-11

Thus we have now expressed the original expression of  xˆ2-8x+5 in the form (x-a)ˆ2-b. Where a = 4 and b = 11.


Glossary

Quadratic Equation - an equation of the form axˆ2+bx+c = 0, where a, b, and c are constants

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