Question

Solve for x:

Answer 1

\[(7-4\sqrt{3})^{x^{2}-4x+3} + (7+4\sqrt{3})^{x^2-4x+3} = 14\]

Answer 2

@ganeshie8

Answer 3

hi
firstly multiply it by
\[(7+4\sqrt{3})^{x^2-4x+3}\]

Answer 4

then let
\[a=(7+4\sqrt{3})^{x^2-4x+3}\]
to get a simple quadratic equation for a and solve for a

Answer 5

just notice that
\[(7-4\sqrt{3})(7+4\sqrt{3})=1\]

Answer 6

how??

Answer 7

what about (7-4root3)^...................?

Answer 8

it is not a simple quadratic eq.

Answer 9

after multiplying it by (7+4root3)^................... u have
\[((7+4\sqrt{3})(7-4\sqrt{3}))^{x^2-4x+3}+(7+4\sqrt{3})^{x^2-4x+3}(7+4\sqrt{3})^{x^2-4x+3}=14(7+4\sqrt{3})^{x^2-4x+3}\]

use my hint to simplify the first term on the left hand side of the equation

Answer 10

can u do that?

Answer 11

@mukushla what eq. you got after replacing it by a.

Answer 12

@ganeshie8

Answer 13

\[(7-4\sqrt{3})^{x^2-4x+3} + (7+ 4\sqrt{3})^{x^2-4x+3} = 14 \]


lets start with @mukushla method... it looks good.

\[(7+ 4\sqrt{3})^{x^2-4x+3} ((7-4\sqrt{3})^{x^2-4x+3} + (7+ 4\sqrt{3})^{x^2-4x+3}]\) = (7+ 4\sqrt{3})^{x^2-4x+3} (14) \]

Answer 14

\[(1)^{x^2-4x+3} + (7+ 4\sqrt{3})^{2(x^2-4x+3)}) = (7+ 4\sqrt{3})^{x^2-4x+3} (14) \]

\[1+ (7+ 4\sqrt{3})^{2(x^2-4x+3)}) = (7+ 4\sqrt{3})^{x^2-4x+3} (14) \]

Answer 15

let a = \((7+ 4\sqrt{3})^{(x^2-4x+3)} \) --------------------(1)
\(1 + a^2 = 14a\) -------------------------------------(2)

we have a qudatratic equation now...
u can try rest as below :
1) solve equation (2) for a
2) equate that with equation (1)
3) take log on both sides.. and solve the FINAL quadratic equation for the value of \(x\)

Enjoy...

Answer 16

@shubham.bagrecha
sorry i lost my connection

Answer 17

@ganeshie8
thank u