Answer to Question #236228 in Math for luka

Question #236228

The roots of the equation 25X2+ X +1 =0 are α^2 and β^2. Find the equation with

integral coefficients whose roots are 1/α and 1/β


1
Expert's answer
2021-09-12T23:52:43-0400
25x2+x+1=025x^2+x+1=0

By Viet Theorem


α2+β2=125\alpha^2+\beta^2=-\dfrac{1}{25}

α2β2=125\alpha^2\beta^2=\dfrac{1}{25}

Then


1α2+1β2=α2+β2α2β2\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}=\dfrac{\alpha^2+\beta^2}{\alpha^2\beta^2}

=125125=1=\dfrac{-\dfrac{1}{25}}{\dfrac{1}{25}}=-1


αβ=15 or αβ=15\alpha\beta=-\dfrac{1}{5}\ \text{or}\ \alpha\beta=\dfrac{1}{5}

αβ=15\alpha\beta=-\dfrac{1}{5}


1α(1β)=5\dfrac{1}{\alpha}(\dfrac{1}{\beta})=-5

(1α+1β)2=1α2+1β2+2αβ(\dfrac{1}{\alpha}+\dfrac{1}{\beta})^2=\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}+\dfrac{2}{\alpha\beta}

=1+2(5)=11=-1+2(-5)=-11

1α+1β=i11 or 1α+1β=i11\dfrac{1}{\alpha}+\dfrac{1}{\beta}=-i\sqrt{11}\text{ or }\dfrac{1}{\alpha}+\dfrac{1}{\beta}=i\sqrt{11}

The quadratic equation is


y2+i11y5=0y^2+i\sqrt{11}y-5=0

Or


y2i11y5=0y^2-i\sqrt{11}y-5=0


αβ=15\alpha\beta=\dfrac{1}{5}

1α(1β)=5\dfrac{1}{\alpha}(\dfrac{1}{\beta})=5

(1α+1β)2=1α2+1β2+2αβ(\dfrac{1}{\alpha}+\dfrac{1}{\beta})^2=\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}+\dfrac{2}{\alpha\beta}

=1+2(5)=9=-1+2(5)=9

1α+1β=3 or 1α+1β=3\dfrac{1}{\alpha}+\dfrac{1}{\beta}=-3\text{ or }\dfrac{1}{\alpha}+\dfrac{1}{\beta}=3


The quadratic equation is


y2+3y+5=0y^2+3y+5=0

Or


y23y+5=0y^2-3y+5=0

y2+i11y5=0y^2+i\sqrt{11}y-5=0


y2i11y5=0y^2-i\sqrt{11}y-5=0


y2+3y+5=0y^2+3y+5=0


y23y+5=0y^2-3y+5=0



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