# CONVERTING FROM STANDARD INTO VERTEX FORM

To convert standard form to vertex form, we may follow the different ways.

## Completing the square method

From the standard form of the equation, y = ax2 + bx + c

(i) Take the coefficient of x2, from all the terms if there is.

(ii)  Write the coefficient of x as a multiple of 2.

(iii)  Get any one of the algebraic identities (a+b)2 or (a-b)2

Example  :

Convert the following quadratics from standard form to vertex form

y = 2x2 – 4x + 8

Solution :

y = 2x2 – 4x + 8

Factor 2.

y = 2(x2 – 2x + 4)

Write the coefficient of x as a multiple of 2.

y = 2(x2 – 2x1 + 1- 12 + 4)

Here x2 – 2x⋅1 + 12 matches with a2 – 2⋅a⋅b + b2 = (a - b)2

y = 2[(x - 1)2 - 1 + 4]

y = 2[(x - 1)2 + 3]

Distributing 2, we get

y = 2(x - 1)2 + 6

## Using short cut

From the standard form of the equation, y = ax2 + bx + c

(i) Take the coefficient of x2, from all the terms if there is.

(ii)  Take half of the coefficient of x and write it as (x - a)2 or (x + a)2. Here a is half the coefficient of x.

Example :

Convert y = x2 - 4x + 3 into factored form.

Solution :

y = x2 - 4x + 3

Coefficient of x2 is 1, so dont have to factor anything.

Half of the coefficient of x is 2.

y = (x - 2)2 - 22 + 3

y = (x - 2)2 - 4 + 3

y = (x - 2)2 - 1

## More Examples

Complete the square to convert the standard form quadratic function into vertex form. Then find the vertex.

Example 1 :

f(x) = x2 + 4x - 14

Solution :

Coefficient of x2 is 1. So, don't have to factorize.

Write the coefficient of x as multiple of 2.

f(x) = x2 + 2 ⋅ ⋅ 2 + 22 - 2 - 14

f(x) = (x+2)2 - 4 - 14

f(x) = (x+2)2 - 18

Example 2 :

f(x) = 2x2 + 9x

Solution :

Coefficient of x2 is 2.

Write the coefficient of x as multiple of 2.

Example 3 :

f(x) = 5x2 - 4x + 1

Solution :

Coefficient of x2 is 5. So, we have to factorize 5.

Example 4 :

f(x) = x2 - 16x + 70

Solution :

Coefficient of x2 is 1. So, don't have to factorize.

Write the coefficient of x as multiple of 2.

f(x) = x2 - 2 ⋅ ⋅ 8 + 82 - 82 + 70

f(x) = (x - 8)2 - 64 + 70

f(x) = (x - 8)2 + 6

Example 5 :

f(x) = -3x2 + 48x - 187

Solution :

Coefficient of x2 is -3. So, factorize -3.

f(x) = -3[x2 - 16x] - 187

Write the coefficient of x as multiple of 2.

f(x) = -3[x2 - 2 ⋅ ⋅ 8 + 82- 82] - 187

f(x) = -3[(x - 8)2- 64] - 187

f(x) = -3(x - 8)2+ 192 - 187

f(x) = -3(x - 8)2+ 5

## Recent Articles

1. ### Graphing Exponential Growth and Decay Worksheet

Dec 08, 23 08:03 AM

Graphing Exponential Growth and Decay Worksheet

2. ### Finding the Specific Term from the Given Explicit Formula Worksheet

Dec 08, 23 07:32 AM

Finding the Specific Term from the Given Explicit Formula Worksheet