Solve the equation (2x-3)(2x-1)(2x+1)(2x+3)=3465 ?
1 Answer
Jun 26, 2017
The solutions are
Explanation:
We start by multiplying out.
We can do this easily by recognizing that
(2x + 3)(2x- 3) = 4x^2 - 9
(2x + 1)(2x- 1) = 4x^2 - 1
(2x - 3)(2x - 1)(2x+ 1)(2x + 3) = (4x^2 - 9)(4x^2 - 1)
(2x- 3)(2x- 1)(2x+ 1)(2x+ 3) = 16x^4 - 36x^2 - 4x^2 + 9
(2x - 3)(2x- 1)(2x+ 1)(2x+ 3) = 16x^4 - 40x^2 + 9
Therefore,
16x^4 - 40x^2 + 9 = 3465
It follows that
16x^4 - 40x^2 - 3456 = 0
2x^4 - 5x^2 - 432 = 0
We now let
2y^2 - 5y - 432 = 0
We can solve by factoring.
2y^2 - 32y + 27y - 432 = 0
2y(y - 16) + 27(y - 16) = 0
(2y + 27)(y - 16) = 0
y = -27/2 and 16
x^2 = -27/2 and 16
x = +- 4 and +- 3sqrt(3/2) i
Hopefully this helps!