# How to obtain a quadratic equation, with integer coefficient, having roots 2+i√5 and 2-i√5 ?

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To obtain a quadratic equation with integer coefficients having roots (2 + i\sqrt{5}) and (2 - i\sqrt{5}), you can use the fact that complex roots occur in conjugate pairs for polynomials with real coefficients.

Given the roots (r_1 = 2 + i\sqrt{5}) and (r_2 = 2 - i\sqrt{5}), the quadratic equation can be expressed as:

[ (x - r_1)(x - r_2) = 0 ]

Expanding this expression, we get:

[ (x - (2 + i\sqrt{5}))(x - (2 - i\sqrt{5})) = 0 ] [ (x - 2 - i\sqrt{5})(x - 2 + i\sqrt{5}) = 0 ]

Now, using the difference of squares, we can simplify this expression:

[ (x - 2)^2 - (i\sqrt{5})^2 = 0 ] [ (x - 2)^2 - (-5) = 0 ] [ (x - 2)^2 + 5 = 0 ]

Expanding further:

[ x^2 - 4x + 4 + 5 = 0 ] [ x^2 - 4x + 9 = 0 ]

So, the quadratic equation with integer coefficients having roots (2 + i\sqrt{5}) and (2 - i\sqrt{5}) is (x^2 - 4x + 9 = 0).

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