Any brilliant Mathematicians here ? i need your help.?

Question

1) this has something do with completing the sq and parabola
x^2 + 3x +1

2x^2 + 4x + 1

2)A room containing 783 square feet is 2 feet longer than it is wide. How long a crown molding is
needed to trim the perimeter of the ceiling?

Answer 1

1) The rule (when the coefficient of x^2 is 1) is

Halve the coefficient of x, square the result, and that's the constant term.
So, you halve 3, square the result.
That means you need
x^2 + 3x + 9/4

but to keep it equal to the original expression you must subtract 9/4 and add 1, thus:
x^2 + 3x +1 = x^2 + 3x + 9/4 - 5/4
which is equal to
(x + 3/2)^2 - 5/4

In the second one, the coefficient of x^2 is 2, so you must write it as
2(x^2 + 2x) + 1
then apply the rule I gave you to x^2 + 2x.

But remember that when you add 1 inside the parentheses, you're really adding 2*1 because of the 2 in front, so then you have to subtract 2*1. Hope you can finish it for yourself.

2) Let the width be x feet, so the length is x + 2. We're told the area is 783, so you should solve

x*(x+2) = 783 -- I'm assuming you've learned that.

One value of x is negative, obviously you use the positive one. The question just means find the perimeter, so, using the value of x you've found, add x to x+2 and double the result.


P.S. 783 = 3*3*3*29, so that should help you find a pair of factors which differ by 2.

Answer 2

competing the square

x^2 + 3x +1 = (x+1.5)^2 - 1.25

2x^2 + 4x + 1 = (sqrt2 x + sqrt2)^2 - 1

2) I have no idea I am a mathematician not a carpenter.

Answer 3

1.)
2x^2 + 4x + 1=y
x^2 + 3x +1=y

2x^2 + 4x + 1=x^2 + 3x +1
x^2+x=0
x(x+1)=0

x=0, x=-1

2.)
x=length
x-2=width

A=length(width)
783 sq ft.=x(x-2)
783 sq ft.=x^2-2x
x^2-2x-783=0
(x-29)(x+27)=0
x=29, x=-27----extraneous

x=29
x-2=27

P=2(27)+2(29)
P=54+58
P=112 ft.

Answer 4

1) y = x^2 + 3x + 1
completing the square we get
y = x^2 + 3x + (3/2)^2 - (3/2)^2 + 1
y = (x + 3/2)^2 - (9/4 -1) = (x+ 3/2)^2 - 5/4
y = ( x + 3/2 - sqrt(5)/2)(x+3/2 + sqrt(5)/2)
y = (x + (3 -sqrt(5))/2)(x + (3 + sqrt(5))/2)

y = 2x^2 + 4x + 1
y/2 = x^2 + 2x + 1/2
y/2 = x^2 + 2x + 1 -1 +1/2
y/2 = (x+1)^2 - 1/2
y/2 = (x+ 1 - sqrt(2)/2) (x+1+sqrt(2)/2)
y = 2(x+(2-sqrt(2))/2)(x+(2+sqrt(2))/2)