Find (2+(sq.root)3)^2 in its simplest form?

Question

I got the answer to the 1st part as 7+4(sq.root)3, but I'm having problems with the second part. it goes...
Hence or otherwise find, in surd, form, the solutions to the equation x^2-28 -16(sq.root)3=0

Answer 1

(a+b)^2=a^2+2ab+b^2

So...a=2 and b=sqrt(3)

(2+sqrt(3))^2 = 2^2 + 2*2*sqrt(3) + sqrt(3)^2

= 4+4sqrt(3)+3=7+4sqrt(3)

x^2-28-16sqrt(3)=0

x^2-4(7+4sqrt(3))=0

x^2 = 4(7+4sqrt(3))

x= +/- 2 sqrt(7+4sqrt(3))

Answer 2

Collect all the terms other than x^2 to the right side,

x^2 = 4(7 +√3) = 4(2+√3)^2

Therefore,

x = ±2(2+√3)

You have to use the conclusion from the first part.

Answer 3

a) (2+(sq.root)3)² = (2+\/3)(2+\/3) =
2² + 2.2.\/3 + (\/3)² = 4 + 4\/3 + 3 = 7 + 4\/3

b) x² - 28 - 16\/3 = 0
x² = (28 + 16\/3)²
x = 28 + 16\/3
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Answer 4

Having done the first part then the second part becomes

x^2 = 28-16root3 by taking other terms than x to the other side

4 is a common factor so

x^2 =4(7+4root3)

so using first part

x^2 =4(2+root3)^2
take sq root of both sides

x = + / – 2(2+root3)

Answer 5

x^2 - 28 - 16sqrt(3) = 0

Bring everything over to the right hand side (other than the x^2).

x^2 = 16sqrt(3) + 28

Now, take the square root of both sides.

x = +/- sqrt ( 16sqrt(3) + 28)

Factor the inside.

x = +/- sqrt (4 [4sqrt(3) + 7])

Pull the 4 out of the square root, which makes it become 2.

x = +/- 2 sqrt (4sqrt(3) + 7)

These are your two solutions:

x = {2 sqrt (4sqrt(3) + 7) , -2 sqrt (4sqrt(3) + 7)}

Answer 6

1st part:
(2+sqrt3)^2=2^2+(sqrt3)^2
+2*2sqrt3
=4+3+4sqrt3
=7+sqrt3
2nd part
now, x^2-28 -16(sq.root)3=0
this gives
x^2= 28+16sqrt3
=4(7+4*sqrt3)
x=+or-{sqrt(4*(7+sqrt3)}
we know from the 1st part
that (2+(sq.root)3)^2
=7+4*sqrt3,hence,
sqrt(7+4sqrt3)=2+sqrt3

therefore,
x=+or-{sqrt(4*(7+4'/sqrt3)}
=+or-{2*sqrt(7+4sqrt3)}
=+or-2*(2+sqrt3)

i hope that this helps

Answer 7

First thing you've got to spot is that

28+16 sq.rt(3) = 4 (7 + 4 sq.rt(3))

So you have

X^2 = [2(2+sq.rt(3))]^2

So

X = plus-or-minus 2(2+sq.rt(3))

Answer 8

here
x = (sq.root)(16(sq.root)3+28)
= 2(sq.root)(4(sq.root)3+7)

Answer 9

+-2(2+(sq.root)3).