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Given the one of the roots for the quadratic equation is: ( 2- 4sqrt3)
Then the other root is: (2+ 4sqrt3)
Since we have the roots we can form a quadratic equation by multiplying the factors:
Let the quadratic function be f(x):
==> f(x) = (x-x1) *(x-x2)
where x1 and x2 are the roots:
==> f(x) = ( x- (2-4sqrt3) ( x- (2+4sqrt3)
= x^2 -(2+4sqrt3)x +(2-4sqrt3)x + (2-4sqrt3)(2+4sqrt3)
= x^2 -2x - 4sqrt3 - 2x +4sqrt x4 - 44
= x^2 - 4x - 44
==> f(x) = x^2 - 4x - 44
To form a quadratic equation with rational coefficients having 2-4(3)^.5 as one of its roots.
Here we are given one root x1 = 2-4(3)^0.5.
Let the other root be x2.
Then we can form the quadratic equation like (x-x1)(x-x2) = 0. Or
x^2 - (x1+x2)x +(x1x2.
Therefore x1+x2= must be rational.
x1x2 = rational.
Therefore we chse the other x2 = 2+3(4)^(0.5) so that
x1+x2 = (2-4(3)^0.5) +(2+4(3)^0.5 = 4 is rational and
x1*x2 = (2-4(3)^0.5) *(2+4(3)^0.5 )= 2^2 - 4^2(3) = -44 is also rational .
Therefore therequired quadratic eqtian is :
x^2 - 4x - 44 = 0.
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