Find the roots of the quadratic equations by using the quadratic formula in each of the following:
(i) \(2x^2 - 3x - 5 = 0\)
(ii) \(5x^2 + 13x + 8 = 0\)
(iii) \(-3x^2 + 5x + 12 = 0\)
(iv) \(-x^2 + 7x - 10 = 0\)
(v) \(x^2 + 2\sqrt{2}\,x - 6 = 0\)
(vi) \(x^2 - 3\sqrt{5}\,x + 10 = 0\)
(vii) \(\dfrac{1}{2}x^2 - \sqrt{11}\,x + 1 = 0\)
(i) \(x=\dfrac{5}{2}\), \(x=-1\)
(ii) \(x=-1\), \(x=-\dfrac{8}{5}\)
(iii) \(x=3\), \(x=-\dfrac{4}{3}\)
(iv) \(x=5\), \(x=2\)
(v) \(x=\sqrt{2}\), \(x=-3\sqrt{2}\)
(vi) \(x=2\sqrt{5}\), \(x=\sqrt{5}\)
(vii) \(x=\sqrt{11}+3\), \(x=\sqrt{11}-3\)
Find the roots of the following quadratic equations by the factorisation method:
(i) \(2x^2 + \dfrac{5}{3}x - 2 = 0\)
(ii) \(\dfrac{2}{5}x^2 - x - \dfrac{3}{5} = 0\)
(iii) \(3\sqrt{2}\,x^2 - 5x - \sqrt{2} = 0\)
(iv) \(3x^2 + 5\sqrt{5}\,x - 10 = 0\)
(v) \(21x^2 - 2x + \dfrac{1}{21} = 0\)
(i) \(x=-\dfrac{3}{2}\), \(x=\dfrac{2}{3}\)
(ii) \(x=-\dfrac{1}{2}\), \(x=3\)
(iii) \(x=\sqrt{2}\), \(x=-\dfrac{\sqrt{2}}{6}\)
(iv) \(x=\dfrac{\sqrt{5}}{3}\), \(x=-2\sqrt{5}\)
(v) double root \(x=\dfrac{1}{21}\)