Convert quadratic function from standard to vertex form

The standard form of a quadratic equation is as follows:
`ax^2 + bx + c = 0`
It's vertex form is as follows:
`a(x- h)^2 + k = 0`
A quadratic equation can be written in both forms. It can be converted from the standard to the vertex form by the method of completing the square.

The following example explains the method.

Consider the quadratic equation in general (standard) form:
`x^2 + 2x + 3 = 0`
We will convert it to vertex form by the following steps:

Move the constant term to the right hand side

The constant term (number without a variable) in the above equation is '3'. We can move it to the right hand side of the = sign as follows:
`x^2 + 2x = -3`

Identify the coefficient of 'x'

The coefficient (number near) 'x' in the above equation is '2'.

Divide the number by 2

Dividing the number '2' by 2, we get 1.

Complete the square

Notice that `2x` in the equation has a `+` sign before it. Thus, we will take the number obtained in the previous step and add it with 'x' and then square the expression as follows:
`(x + 1)^2`
Now we can replace the left hand side of the quadratic equation with the above expression as follows:
`(x + 1)^2 = -3`

Move the constant term in the right hand side again to the left hand side of the equation

`(x + 1)^2 + 3 = 0`
The above equation is said to be the vertex form of `x^2 + 2x + 3 = 0`

No comments:

Post a Comment