How to Re-write completing the square: x2+x+1

Kinsley Moon

Kinsley Moon

Answered question


How to Re-write completing the square:

Answer & Explanation

Darrell Boone

Darrell Boone

Beginner2022-01-21Added 9 answers

Step 1
Remember the formula for the square of a binomial:
Now, when you see x2+x+1, you want to think of x2+x as the first two terms you get in expanding the binomial (x+c)2 for some c; that is,
Since the middle term should be 2cx, and you have x, that means that you want 2c=1, or c=12.
But if you have (x+12)2, you get x2+x+14. Since all you have is x2+x, you complete the square by adding the missing 14.
Since you are not allowed to just add constants willy-nilly, you must also cancel it out by subtracting 14. So:
=(x2+2(12)x+)+1 figuring out what c is
=(x2+2(12)x+(12)2)14+1 completing the square


Beginner2022-01-22Added 10 answers

Step 1
One can rewrite a degree n>1 polynomial f(x)=xn+bxn1+ into a form such that its two highest degree terms are ''absorbed'' into a perfect n'th power of a linear polynomial, namely
has degree n2
When n=2 this is called completing the square - esp. when used to solve a quadratic equation. If g(x)=g is constant (as is always true when n=2) then this yields a closed form for the roots of f(x), namely




Skilled2022-01-27Added 437 answers

Step 1 As Arturo points out what you have to observe is the coefficient. x2+x+1=x2+x+1+34 =x2+x+14+34 =(x+12)2+34 I am sure once you get used to such type of things you shall not have trouble in doing such problems. Solve more problems based on this type. Suppose you have the coefficient of x as a note that a24 should be added and subtracted from the constant term. What i mean by this is: Suppose you have something of this type x2+ax+b2 then you can write this as (x+a2)2+b2a24.

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