Question

Please help :)
Find the rational values of a and b with a > 0 by forming 2 simultaneous equations and solving them by substitution. (part d)

Answer 1

\[(a + b \sqrt{7})^{2} = \frac{ 37 }{ 4 } + 3\sqrt{7}\]

Answer 2

it would be \[a^{2} +2ab \sqrt{7} + 7b ^{2} = \frac{ 37 }{ 4 } + 3\sqrt{7}\]

Answer 3

i can get the 2 simultaneous equations but i cant go any further
\[a^{2} + 7b ^{2} = \frac{ 37 }{ 4 }\]
\[2ab \sqrt{7} = 3\sqrt{7}\]

Answer 4

Wolfram|Alpha http://www.wolframalpha.com/input/?i=solve+%28a+%2B+b+%5Csqrt%7B7%7D%29%5E%7B2%7D+%3D+%5Cfrac%7B+37+%7D%7B+4+%7D+%2B+3%5Csqrt%7B7%7D+

didn't give me a solution. Not sure why. Is \((a + b \sqrt{7})^{2} = \frac{ 37 }{ 4 } + 3\sqrt{7}\) all you're given?

Answer 5

yea im sure thats all im given

Answer 6

anyways answers were b = 1
a = \[\frac{ 3 }{ 2 }\]

Answer 7

@Kuoministers you've done correctly thus far.

Answer 8

Why dont you square both sides of the 2nd eqn.
You'll then get
4a^2 b^2 = 9

after that, let a^2 = x and y^2 = b

See if that helps ?

Answer 9

And yes, thats the right answer.

Answer 10

Never seen questions like this before, and W|A http://www.wolframalpha.com/input/?i=a%5E%7B2%7D+%2B+7b+%5E%7B2%7D+%3D+%5Cfrac%7B+37+%7D%7B+4+%7D%2C+2ab+%5Csqrt%7B7%7D+%3D+3%5Csqrt%7B7%7D gave me 4 solutions. Can you explain a bit how you got that answer @shubhamsrg ? Thanks.

Answer 11

Wolfram gave 4 solutions, but we are required to choose only those solutions where a>0 and is rational

+ on comparison, we have 2 eqns as @Kuoministers rightly formed.
so 2 eqns, 2 variables, easy to solve.

Answer 12

im sorry but squaring it doesn't seem to help?

Answer 13

after squaring 2nd eqn,
you get 4a^2 b^2 =9 right?
Now let a^2 = x and b^2 =y

Your 2 eqns will be modified as

x + 7y = 37/4
and
4xy = 9

Following ?

Answer 14

yes

Answer 15

Now you have 2 eqns and 2 variables, you should be able to solve now hmm?

Answer 16

ok gimme a min ill try

Answer 17

i cant get the right answer... idk why...
im using substitution method

Answer 18

\[x = \frac{ 9 }{ 4y }\]

Answer 19

\[\frac{ 9 }{ 4y } + 7y = \frac{ 37 }{ 4 }\]

Answer 20

Correct so far. Please proceed.

Answer 21

\[\frac{ 9 }{ y } + 28y = 37\]

Answer 22

\[9 + 28y ^{2} = 37y\]

Answer 23

Its 37/4 there*

Answer 24

i multiplied the entire equation by 4

Answer 25

and y

Answer 26

ohh sorry,my bad :P
please continue..

Answer 27

\[28y ^{2} - 37y + 9 = 0\]

Answer 28

yes, correct.

Answer 29

split 37 as 28+ 9

Answer 30

(28y - 9)(y - 1)

Answer 31

btw why wont y = \[\frac{ 9 }{ 28 }\]

Answer 32

y = b^2
b is rational right ?
so y cant take value 9/28 since that'll make b irrational.

Answer 33

your right thanks so much

Answer 34

hmm, glad i could help..