Confusing question

1

12345678

Junior Member
Joined
Mar 30, 2013
Messages
102
‘Find the constants a and b such that, for all values of x, x² + 6x + 20 = (x + a)² + b. Hence state the least values of x² + 6x + 20, and state also the value of x for which this least value occurs’.
I understand the start of the question: it is just completing the square.
X² + 6X +20 = (X + 3)² - 9 + 20 = (X + 3)² +11.
Therefore, a=3 and b=11.
However, I am unsure as to what the second part of the question (Hence state…) is actually asking me to do. If anybody could explain what it is asking, it would be appreciated.
 
"x^2+ 6x+ 9" doesn't look like anything! "y= x^2+ 6x+ 9" and "y= (x+ 3)^2" have graphs. If you let u= x+ 3, (so u= 0 when x= -3) that becomes "y= u^2". Do you know what the graph of "y= x^2" looks like?
 
HallsofIvy said:
"x^2+ 6x+ 9" doesn't look like anything! "y= x^2+ 6x+ 9" and "y= (x+ 3)^2" have graphs. If you let u= x+ 3, (so u= 0 when x= -3) that becomes "y= u^2". Do you know what the graph of "y= x^2" looks like?
Click to expand...
Its a U shaped curve that has symmetry about the y axis
 
Punch-Line

For y(x) = (x + a)² + b
The minimum of y(x) is at (x,y) = (-a, b)

That is why your problem suggests that you write y(x) in that form.
 
You are going out of your way to make this problem as hard as possible! Finding a minimum value for \(\displaystyle x^2+ 6x+ 20= x^2+ 6x+ 9+ 11= (x+ 3)^2+ 11\) does NOT require any calculus at all. This is purely an algebra problem. You have been asked repeatedly if you know what the graph of "\(\displaystyle y= x^2\)" looks like. Do you? What is the minimum value of \(\displaystyle y= x^2\)? If you can do that, what about \(\displaystyle y= x^2+ 11\)? And then \(\displaystyle y= (x+ 3)^2+ 11\)?
 
Last edited:
HallsofIvy said:
You are going out of your way to make this problem as hard as possible! Finding a minimum value for \(\displaystyle x^2+ 6x+ 20= x^2+ 6x+ 9+ 11= (x+ 3)^2+ 11\) does NOT require any calculus at all. This is purely an algebra problem. You have been asked repeatedly if you know what the graph of "\(\displaystyle y= x^2\)" looks like. Do you? What is the minimum value of \(\displaystyle y= x^2\)? If you can do that, what about \(\displaystyle y= x^2+ 11\)? And then \(\displaystyle y= (x+ 3)^2+ 11\)?
Click to expand...
The minimum value of X^2 is (0,0), y = x^2 + 11, the min value is (0,11) and (X+3)^2 + 11, the min value is (-3,11). Note: Why does the answer book give the minimum value as (11,-3)?
 
Last edited:
12345678 said:
The minimum value of X^2 is (0,0), y = x^2 + 11, the min value is (0,11) and (X+3)^2 + 11, the min value is (-3,11). Note: Why does the answer book give the minimum value as (11,-3)?
Click to expand...

Excellent! I would say the minimum value of \(\displaystyle x^2\) is 0 which occurs at x= 0. Similarly the minimum value of \(\displaystyle x^2+ 11\) is 11 which occurs at x= 0 and the minimum value of \(\displaystyle (x+ 3)^2+ 11\) is 11 which occurs at x= -3. I have no idea why an answer book would give (11, -3), in particular because that is a point, not a value.
 
12345678 said:
Why does the answer book give the minimum value as (11,-3)?
Click to expand...
Perhaps because the problem first asked for minimum (which is 11) and secondly asked at what x value (which is -3).
But it is unfortunate that the ordered answer set isn't written as {11, -3}
 
You must log in or register to reply here.
Top