the roots of 2x^2-3x-2=0 are alpha and beta find an equation with integral coefficients whose roots are 2alpha+beta and 2beta+alpha

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2x^2-3x-2=0

First let us find the zeros of the equation:

(2x+1)(x-2)=0

==> x= -1/2 and x=2

Or alpha=-1/2  and beta=2

Now we need to find the equation whose roots are:

2alpha+beta or 2(-1/2)+2=1

and 2beta+alpha  OR  2(2)-1/2 = 7/2

Then the factors are:

(x-1)(x-7/2)=0

 x^2-(9/2)x+7/2=0

Multiply by 2:

2x^2+9x+7=0

Then the...

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2x^2-3x-2=0

First let us find the zeros of the equation:

(2x+1)(x-2)=0

==> x= -1/2 and x=2

Or alpha=-1/2  and beta=2

Now we need to find the equation whose roots are:

2alpha+beta or 2(-1/2)+2=1

and 2beta+alpha  OR  2(2)-1/2 = 7/2

Then the factors are:

(x-1)(x-7/2)=0

 x^2-(9/2)x+7/2=0

Multiply by 2:

2x^2+9x+7=0

Then the function is:

f(x) = 2x^2+9x+7

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