Question

Complete the square to determine the maximum or minimum value of the function defined by the expression. x2 + 30x + 21 A) maximum value at 15 B) maximum value at 30 C) maximum value at −15 D) minimum value at −204

Answer 1

x^2 + 30x + 21
= ( x + 15)^2 - 225 + 21
= (x + 15)^2 - 204

Its D minimum at -204

Answer 2

`x^2 + 30x + 21=(x+15)^2-204`

Option D - minimum value at −204

Explanation:

Given : Expression
`x^2 + 30x + 21`

To find : Complete the square to determine the maximum or minimum value of the function defined by the expression?

Solution :

The general form of quadratic equation is
`ax^2+bx+c=0`

To convert into complete square the form is
`a(x+d)^2+e=0`

Where,
`d=(b)/(2a)` and
`e=c-(b^2)/(4a)`

Now, comparing the given equation
`x^2 + 30x + 21`

a=1 , b=30, c=21

`d=(b)/(2a)=(30)/(2(1))=15`

`e=21-(30^2)/(4(1))=21-225=-204`

Substitute in
`a(x+d)^2+e=0`

`1(x+15)^2+(-204)=0`

`(x+15)^2-204=0`

`[tex]x^2 + 30x + 21`[/tex]

By Completing the square we get,

`x^2 + 30x + 21=(x+15)^2-204`

The minimum value of this is when x+15 = 0 ⇒ x=-15

i.e,
`(-15+15)^2-204=-204`

Therefore, Option D is correct.

Minimum value of the function is at -204.