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Properties of Determinants

If a and b ar...

Question

If a and b are rational and α and β be the roots of x^2+2ax+b=0, then the equation with rational coefficients one of whose roots is α+β+\sqrt{α^2+β^2}

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a

b

are rational and

α, β

be the roots of

x^{2} + 2 a x + b = 0,

then the equation with rational coefficients one of whose roots is

α + β + \sqrt{α^{2} + β^{2}}

α, β

are the roots of the equation

x^{2} - a x + b = 0

and roots of the eqaution

b x^{2} - 4 x + 4 = 0

α + \frac{β^{2}}{α}

β + \frac{α^{2}}{β}

a + b

α \neq β

α^{2} = 5 α - 3

β^{2} = 5 β - 3

then the equation whose roots are

α / β

β / α

α \neq β

α^{2} = 2 α - 3; β^{2} = 2 β - 3

, then the equation whose roots are

\frac{α}{β}

\frac{β}{α}

α

β

are the roots of the equation

x^{2} - 4 \sqrt{2} k x + 2 e^{4 l n k} - 1 = 0

k

α^{2} + β^{2} = 66

α^{2} + β^{2} + α + β

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