Vertex form of a quadratic equation
We want to convert the following equation:
y = x² + 2x + 50
into the vertex form.
Step 1: completing the square
We know that the perfect square trinomial is given by:
(x + a)² = x² + 2 · a · x + a²
In this case, we are going to take into account the first and second term of the equation:
y = x² + 2x + 50
Since, we can write the equation as:
y = x² + 2 · 1 · x + 50
then, if we wanted to obtain
(x + 1)² = x² + 2 · 1 · x + 1² = x² + 2x + 1
then we will need a 1:
y = x² + 2x + 50
↓
y = x² + 2 · 1 · x + 1 - 1 + 50
Then,
y = (x² + 2 · 1 · x + 1) - 1 + 50
↓
y = (x + 1)² - 1 + 50
↓
y = (x + 1)² + 49
Vertex
Given the following equation:
y = (x - h)² + k,
we know then that the vertex is at
(h, k)
In this case
y = (x + 1)² + 49
↓
y = (x - (-1))² + 49
h = -1 and k = 49
Then
(h, k) = (-1, 49)
Answer- The equation in vertex form is: y = (x + 1)² + 49 and its vertex is at (-1, 49)