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Algebra

Question 8 of 20

Surds

Question

Express \(\left(\sqrt{6} - 2\sqrt{3}\right)^{2}\) in the form \(a + b\sqrt{c}\)

Show solution Hide solution Fully worked — 5 steps
  1. Expand the square

    Since \((x - y)^{2} = x^{2} - 2xy + y^{2}\), where \(x = \sqrt{6}\) and \(y = 2\sqrt{3}\):

    \[ \left(\sqrt{6} - 2\sqrt{3}\right)^{2} = \left(\sqrt{6}\right)^{2} - 4\left(\sqrt{6}\right)\left(\sqrt{3}\right) + \left(2\sqrt{3}\right)^{2} \]
  2. Multiply the surds

    Using the rule \(\sqrt{a} \times \sqrt{b} = \sqrt{(a \times b)}\):

    \[ = 6 - 4\sqrt{6 \times 3} + 12 \]
  3. Add the integers
    \[ = 18 - 4\sqrt{18} \]
  4. Split \(\sqrt{18}\) into perfect-square factors

    Using the rule \(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\):

    \[ = 18 - 4\left(\sqrt{9} \times \sqrt{2}\right) \]
  5. Final answer
    \[ \left(\sqrt{6} - 2\sqrt{3}\right)^{2} = 18 - 12\sqrt{2} \]

    That is, the simplified expression is eighteen minus twelve times the square root of two, which is in the required form \(a + b\sqrt{c}\).