Simple question - completing square!!!?

Question

Hi.. I'm really not understanding how to complete the square (especially when there are negatives). This is an A Level C1 past exam qu, and I really have no idea how to do it. I only presume that I need to use completing the square, but another method would be fine.

Please show your full method:

f(x) = 9 +6x -x^2

i) Find the values of A and B such that
f(x) = A - (x+B)^2

ii) State the maximum value of f(x)


Thank you vv much =D

Answer 1

you got a CI exam in January too? good luck =]

right. so f(x) = -x^2 +6x +9 (rearranged)

to complete the square you would get f(x) = -(x-3)^2 +9+9

3 is inside the bracket as it's half the coefficient (i.e. half of 6 - 6 is the coefficient of x). And in most cases of completing the square we've come across so far in maths, we put a negative number outside the brackets (this case 9). But for this one we put a positive 9 outside the brackets as the outcome of +3 multiplied by -3 gives -9 therefore it has to be a positive 9 to cancel this out.

therefore we get:

f(x) = -(x-3)^2 + 18

compare the above to f(x) = A - (x+B)^2

A = 18 and B = -3



ii) to get the maximum value of f(x), the x in the equation needs to be the smallest number possible (i can never remember why though) which is 0 here (and in most cases i think). It can't be a negative number as -(x-3)^2 will always be positive since it is being squared.

so if x is 0 then you are left with f(x) = 18 therefore 18 is the maximum value of f(x)



hope that helped!

Answer 2

PART I:

The easiest way would be to work backwards by taking your final form and expanding it out.

You are trying to get to:
f(x) = A - (x + B)²

Expanding this you get:
f(x) = A - (x² + 2Bx + B²)

Distributing the - sign through:
f(x) = -x² - 2Bx - B² + A

Compare this to your original equation:
f(x) = -x² + 6x + 9

Since the coefficient on x is 6, you can equate:
-2Bx = 6x
-2B = 6
B = -3

And your last coefficient is 9 which is the same as:
- B² + A = 9

-(-3)² + A = 9
-9 + A = 9
A = 18

Alternatively, you can do this working forward.

You start with:
f(x) = 9 - x² + 6x

Pull out the negative sign from the x² and x terms:
f(x) = 9 - (x² - 6x)

To complete the square, take the coefficient on x (-6), halve it (-3) and square it (9). Add and subtract this value inside the parentheses:

f(x) = 9 - (x² - 6x + 9 - 9)

Now take the -9 outside the parentheses (don't forget the sign change):
f(x) = 9 - (x² - 6x + 9) + 9

Combine the two 9s:
f(x) = 18 - (x² - 6x + 9)

The last part in the parentheses is now a perfect square:
f(x) = 18 - (x - 3)²

PART II:

The squared term (x - 3)² can't be negative. The smallest value that you can subtract is zero (when x = 3). That results in 18 - 0, or a maximum value of 18.

Answer 3

Isolate the x words: x^2 + 8x = 4 a million/2 the fee of 8x, that provides you with 4. Now sq. that and you will get sixteen, so upload sixteen to the two facets: x^2 + 8x + sixteen = 20 ingredient it out: (x + 4)(x + 4) = 20 as the two contraptions of brackets are same, integrate them right into a sq.: (x + 4)^2 = 20 sq. root the two facets: x + 4 = ± ?20 Subtract 4 from the two facets: x = -4 ± ?20 Simplify the unconventional: x = -4 ± 2?5

Answer 4

I'm gonna tell you how I do it in my head. (might not work for you)

change the equation so that the x sq is positive and the 6x is on the same side as the x sq, leaving the constant (9) on the other side. we are worried about the side with x right now. divide 6 by two and square it to make 9. then you add 9 on the other side of the equation.

you know what I give up. im gonna sleep

Answer 5

i)
f(x) = 9+6x-x²
3²+9-(x²-6x+3²)
18-(x-3)²
f(x) = A - (x+B)²
A - (x+B)² = 18 - (x-3)²

A=18 , B=-3

ii) Max value = 18 .

Answer 6

f(x) = 9 + 6x -- x^2 = 18 -- (x --3)^2 whence A = 18, B = -- 3
maximum value of f(x) is f(3) = 18

Answer 7

Part (i)
- ( x ² - 6x - 9 )
- ( x ² - 6x + 9 - 9 - 9 )
- [ (x - 3) ² - 18 ]

f(x) = 18 - (x - 3) ²
A = 18 , B = - 3

Part (ii)
Max. value of f(x) = 18