Concept:For a quadratic equation of the form:ax2 “ bx + c = 0, the roots will be given by:\(x = \frac{{b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\) For the roots to be equal, b2 “ 4ac should equal 0, i.e.b2 = 4ac ---(1)Given that the value of the equal root is ?, i.e.\(\beta = \frac{b}{{2a}}\) ---(2)From the above condition, we conclude that option (1) is not correct.Squaring equation (2), we get:\({\beta ^2} = \frac{{{b^2}}}{{{{\left( {2a} \right)}^2}}}\) Putting b2 from equation (1) in the above, we get:\({\beta ^2} = \frac{{4ac}}{{{{\left( {2a} \right)}^2}}} = \frac{{4ac}}{{4{a^2}}}\)\({\beta ^2} = \frac{c}{a}\)? option (2) is also incorrect.Cubing equation (2), we can also write:\({\beta ^3} = \frac{{{b^3}}}{{{{\left( {2a} \right)}^3}}}\) With b2 = 4acb3 = 4abc\(\therefore {\beta ^3} = \frac{{4abc}}{{{{\left( {2a} \right)}^3}}}\)\({\beta ^3} = \frac{{bc}}{{2{a^2}}}\)? Option (3) is correct.Option (4) cannot be correct, because for the root to be equal, b2 = 4ac must satisfy.