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Help me with this one

Help me with this one-example-1
User DamienL
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Answer:

"complete the square" to put in vertex form

Explanation:

It may be helpful to consider the square of a binomial:

(x +a)² = x² +2ax +a²

The expression x² +x +1 is in the standard form of the expression on the right above. Comparing the coefficients of x, we see ...

2a = 1

a = 1/2

That means we can write ...

(x +1/2)² = x² +x +1/4

But we need x² +x +1, so we need to add 3/4 to the binomial square in order to make the expressions equal:


x^2+x+1\\\\=(x^2+x+(1)/(4))+(3)/(4)\\\\=(x+(1)/(2))^2+(3)/(4)

_____

Another way to consider this is ...

x² +bx +c

= x² +2(b/2)x +(b/2)² +c -(b/2)² . . . . . . rewrite bx, add and subtract (b/2)²*

= (x +b/2)² +(c -(b/2)²)

for b=1, c=1, this becomes ...

x² +x +1 = (x +1/2)² +(1 -(1/2)²)

= (x +1/2)² +3/4

_____

* This process, "rewrite bx, add and subtract (b/2)²," is called "completing the square"—especially when written as (x-h)² +k, a parabola with vertex (h, k).

User Annibigi
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