Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

Write whether the following statements are true or false. Justify your answers.

(i) Every quadratic equation has exactly one root.

(ii) Every quadratic equation has at least one real root.

(iii) Every quadratic equation has at least two roots.

(iv) Every quadratic equation has at most two roots.

(v) If the coefficient of x² and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.

(vi) If the coefficient of x² and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.

A quadratic equation with integral coefficient has integral roots. Justify your answer.

Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rational? Justify.

Is 0.2 a root of the equation x² - 0.4 = 0? Justify your answer.