How to complete the square y=x^2+4x+1 . As well as find the vertex

Question

I have to complete the square and determine the vertex. 

Answer 1

y = x² + 4x                 + 1

y = x² + 4x + 2²          + 1 - 2²

y = (x + 2)² + 1 - 4

y = (x + 2)² - 3            in y = (x - h)² + k, (h, k) is the vertex.

 

vertex: (-2, -3)

Answer 2

Parabolic equations can be written in the form:

 

y= A(x-h)^2+k

 

Since you have a MONIC polynomial (leading coefficient, of x^2 is 1), A=1 in the formula above.

 

The beauty of writing a quadratic or parabola this way, is that the vertex (the minimum y value if A>0; or maximum y when A is negative) occurs at the point (h,k) in the xy-plane.

 

To convert x^2+bx+c into this form, we use the technique of completing the square

 

Since (x-h)^2+k = (x-h)(x-h) + k = x^2-2hx+h^2+k,

 

we look to find (h,k) such that the coefficients match with the initial equation, that is to say we want

-2h=b, and h^2+k=c

so we first set h=-b/2 and recognize that we are adding and subtracting (-b/2)^2 to "complete the square"

Using this value for h, we can find k

(-b/2)^2+k=c, so k=c-(-b/2)^2...

 

Thus the quadratic x^2+bx+c = (x-b/2)^2+(c-(-b/2)^2), and has vertex located at the point (-b/2, c-(-b/2)^2) in the xy-plane.

 

Your problem above starts with b=4 and c=1.

 

Hope that helps!

Answer 3

To complete the square, we put the function in vertex form.

 

y = a(x - h)² + k

 

where:

a is the coefficient of x² term

 

Vertex has coordinates (h, k)

 

 

y = (x² + 4x + 4) - 3

 

y = (x + 2)(x + 2) - 3

 

y = (x + 2)² - 3

 

 

h = -2

k = -3

 

Vertex is (-2, -3)