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

Answer:

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

Option D - minimum value at −204

Step-by-step 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, $d=\frac{b}{2a}$ and $e=c-\frac{b^2}{4a}$

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

a=1 , b=30, c=21        

$d=\frac{b}{2a}=\frac{30}{2(1)}=15$

$e=21-\frac{30^2}{4(1)}=21-225=-204$

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.