Solution:

 \(x^{2} - 15x + 54 = 0\)

∴ \(\underline {x^{2} - 9x}\ \underline {-\ 6x + 54} = 0\)

∴ \(x\underline {(x - 9)}\ - 6 \underline {(x - 9)} = 0\)

∴ \((x - 9)(x - 6) = 0\)

∴ \(x - 9 = 0 \text { OR } x - 6 = 0\)

∴ \(x = 9 \text { OR } x = 6\)

∴ 9, 6 are the roots of the given quadratic equation.

Solution:

 \(x^{2} + x - 20 = 0\)

∴ \(\underline {x^{2} + 5x}\ \underline {-\ 4x -\ 20} = 0\)

∴ \(x\underline {(x + 5)}\ - 4 \underline {(x + 5)} = 0\)

∴ \((x + 5)(x - 4) = 0\)

∴ \(x + 5 = 0 \text { OR } x - 4 = 0\)

∴ \(x = - 5 \text { OR } x = 4\)

∴ − 5, 4 are the roots of the given quadratic equation.

Solution:

 \(2y^{2} + 27y + 13 = 0\)

∴ \(\underline {2y^{2} + 26y}\ \underline {+ y + 13} = 0\)

∴ \(2y\underline {(y + 13)}\ + 1 \underline {(y + 13)} = 0\)

∴ \((y + 13)(2y + 1) = 0\)

∴ \(y + 13 = 0 \text { OR } 2y + 1 = 0\)

∴ \(y = - 13 \text { OR } 2y = - 1\)

∴ \(\displaystyle y = - 13 \text { OR } y = - \frac {1}{2}\)

∴ \(\displaystyle - 13, - \frac {1}{2}\) are the roots of the given quadratic equation.

Solution:

 \(5m^{2} = 22m + 15\)

∴ \(5m^{2} - 22m - 15 = 0\)

∴ \(\underline {5m^{2} - 25m}\ \underline {+ 3m - 15} = 0\)

∴ \(5m\underline {(m - 5)}\ + 3 \underline {(m - 5)} = 0\)

∴ \((m - 5)(5m + 3) = 0\)

∴ \(m - 5 = 0 \text { OR } 5m + 3 = 0\)

∴ \(m = 5 \text { OR } 5m = - 3\)

∴ \(\displaystyle m = 5 \text { OR } m = - \frac {3}{5}\)

∴ \(\displaystyle 5, - \frac {3}{5}\) are the roots of the given quadratic equation.

Solution:

 \(\displaystyle 2x^{2} - 2x + \frac {1}{2} = 0\)

Multiplying both sides by 2,

 \(\displaystyle 2x^{2} \times {2} - 2x \times {2} + \frac {1}{\cancel {2}} \times {\cancel {2}} = 0 \times {2}\)

∴ \(4x^{2} - 4x + 1 = 0\)

∴ \(\underline {4x^{2} - 2x}\ \underline {-\:2x + 1} = 0\)

∴ \(2x\underline {(2x - 1)}\ - 1 \underline {(2x - 1)} = 0\)

∴ \((2x - 1)(2x - 1) = 0\)

∴ \(2x - 1 = 0 \text { OR } 2x - 1 = 0\)

∴ \(2x = 1 \text { OR } 2x = 1\)

∴ \(\displaystyle x = \frac {1}{2} \text { OR } x = \frac {1}{2}\)

∴ \(\displaystyle \frac {1}{2}, \frac {1}{2}\) are the roots of the given quadratic equation.

Solution:

 \(\displaystyle 6x - \frac {2}{x} = 1\)

Multiplying both sides by x,

 \(\displaystyle 6x \times {x} - \frac {2}{\cancel {x}} \times {\cancel {x}} = 1 \times {x}\)

∴ \(6x^{2} - 2 = x\)

∴ \(6x^{2} - x - 2 = 0\)

∴ \(\underline {6x^{2} - 4x}\ \underline {+\ 3x - 2} = 0\)

∴ \(2x\underline {(3x - 2)}\ + 1 \underline {(3x - 2)} = 0\)

∴ \((3x - 2)(2x + 1) = 0\)

∴ \(3x - 2 = 0 \text { OR } 2x + 1 = 0\)

∴ \(3x = 2 \text { OR } 2x = -\:1\)

∴ \(\displaystyle x = \frac {2}{3} \text { OR } x = -\:\frac {1}{2}\)

∴ \(\displaystyle \frac {2}{3}, -\:\frac {1}{2}\) are the roots of the given quadratic equation.

Solution:

 \(\displaystyle \sqrt {2}x^{2} + 7x + 5\sqrt {2} = 0\)

∴ \(\displaystyle \sqrt {2}x^{2} + \bbox[white, 5pt, border: 3px solid red]{5x} + \bbox[white, 5pt, border: 3px solid red]{2x} + 5\sqrt {2} = 0\)

∴ \(\displaystyle x\left (\bbox[white, 5pt, border: 3px solid red]{\sqrt {2}x + 5}\right) + \sqrt {2}\left (\bbox[white, 5pt, border: 3px solid red]{\sqrt {2}x + 5}\right) = 0\)

∴ \(\displaystyle x\left (\bbox[white, 5pt, border: 3px solid red]{\sqrt {2}x + 5}\right)\left (x + \sqrt {2}\right) = 0\)

∴ \(\displaystyle x\left (\bbox[white, 5pt, border: 3px solid red]{\sqrt {2}x + 5}\right) = 0 \text { OR }\left (x + \sqrt {2}\right) = 0\)

∴ \(\displaystyle x = \bbox[white, 5pt, border: 3px solid red]{- \frac {5}{\sqrt {2}}} = 0 \text { OR }x = - \sqrt {2}\)

∴ \(\displaystyle \bbox[white, 5pt, border: 3px solid red]{- \frac {5}{\sqrt {2}}} \text { and } - \sqrt {2}\) are roots of the equation.

Solution:

 \(\displaystyle 3x^ {2} - 2\sqrt {6}x + 2 = 0\)

∴ \(\underline {3x^{2} - \sqrt {6}x}\ \underline {-\ \sqrt {6}x + 2} = 0\)

∴ \(\sqrt {3}x\underline {(\sqrt {3}x - \sqrt {2})}\ - \sqrt {2} \underline {(\sqrt {3}x - \sqrt {2})} = 0\)

∴ \((\sqrt {3}x - \sqrt {2})(\sqrt {3}x - \sqrt {2}) = 0\)

∴ \(\sqrt {3}x - \sqrt {2} = 0 \text { OR } \sqrt {3}x - \sqrt {2} = 0\)

∴ \(\sqrt {3}x = \sqrt {2} \text { OR } \sqrt {3}x = \sqrt {2}\)

∴ \(\displaystyle x = \frac {\sqrt {2}}{\sqrt {3}} \text { OR } x = \frac {\sqrt {2}}{\sqrt {3}}\)

∴ \(\displaystyle \frac {\sqrt {2}}{\sqrt {3}},\ \frac {\sqrt {2}}{\sqrt {3}}\) are the roots of the given quadratic equation.

Solution:

 \(2m(m - 24) = 50\)

∴ \(2m^ {2} - 48m = 50\)

∴ \(2m^ {2} - 48m - 50 = 0\)

Dividing both sides by 2,

∴ \(m^ {2} - 24m - 25 = 0\)

∴ \(\underline {m^{2} - 25m}\ \underline {+\ 1m -\:25} = 0\)

∴ \(m\underline {(m - 25)}\ + 1\underline {(m - 25)} = 0\)

∴ \((m - 25)(m + 1) = 0\)

∴ \(m - 25 = 0 \text { OR } m + 1 = 0\)

∴ \(m = 25 \text { OR } m = - 1\)

∴ \(25, - 1\) are the roots of the given quadratic equation.

Solution:

 \(25m^{2} = 9\)

∴ \(25m^{2} - 9 = 0\)

∴ \((5m)^{2} - (3)^{2} = 0\)

∴ \((5m + 3)(5m - 3) = 0\)
  \(\left[\because a^2 - b^2 = (a + b)(a - b)\right]\)

∴ \(5m + 3 = 0 \text { OR } 5m - 3 = 0\)

∴ \(5m = - 3 \text { OR } 5m = 3\)

∴ \(\displaystyle m = - \frac {3}{5} \text { OR } m = \frac {3}{5}\)

∴ \(\displaystyle - \frac {3}{5}, \frac {3}{5}\) are the roots of the given quadratic equation.

Solution:

 \(7m^{2} = 21m\)

∴ \(7m^{2} - 21m = 0\)

∴ \(7m(m - 3) = 0\)

∴ \(7m = 0 \text { OR } m - 3 = 0\)

∴ \(m = 0 \text { OR } m = 3\)

∴ 0, 3 are the roots of the given quadratic equation.

Solution:

 \(m^{2} - 11 = 0\)

∴ \(m^{2} - \left(\sqrt {11}\right)^{2} = 0\)

∴ \(\left(m + \sqrt {11}\right)\left(m - \sqrt {11}\right) = 0\)

  \(\left[\because a^2 - b^2 = (a + b)(a - b)\right]\)

∴ \(\left(m + \sqrt {11}\right) = 0 \text { OR } \left(m - \sqrt {11}\right) = 0\)

∴ \(m = -\:\sqrt {11} \text { OR } m = \sqrt {11}\)

∴ \(-\:\sqrt {11}, \sqrt {11}\) are the roots of the given quadratic equation.