Number system practice

Rationalise & find value of cubic expression

Question: If x = \\frac{1}{8-\sqrt{60}}\\), what is the value of \x^{3}-5x^{2}+8x-4?\\)

Video Explanation

Explanatory Answer

In all these questions, before plugging the value of x in the expression whose value is to be computed, rationalize x and get x expressed with a rational denominator.
You are likely to find a clue after this step.

Step 1: Rationalise \\frac{1}{8 - \sqrt{60}}\\)

\\frac{1}{8 - \sqrt{60}}\\) x \\frac{8 + \sqrt{60}}{8 + \sqrt{60}}\\)

=\\frac{8 + \sqrt{60}}{4}\\) = 2 + \\frac{\sqrt{15}}{2}\\)

Or x - 2 = \\frac{\sqrt{15}}{2}\\)

Step 2: Rewrite \x^3 - 5x^2 + 8x - 4\\)

Because x = 2 + \\frac{\sqrt{15}}{2}\\), try and express \x^3 - 5x^2 + 8x - 4\\) in terms of \{(x-2)}^3, {(x-2)}^2\\) and \x - 2\\).
\{(x - 2)}^3 = x^3 - 6x^2 + 12x - 8\\) -------- (1)
\{(x - 2)}^2 = x^2 - 4x + 4\\) -------- (2)

Add or subtract the two equations to check whether you get the given expression

(1) + (2) = \x^3 - 5x^2 + 8x - 4\\)
So, \x^3 - 5x^2 + 8x - 4\\) = \{(x-2)}^3\\) + \{(x-2)}^2\\)
=\{(\frac{\sqrt{15}}{2})^3}\\) + \{(\frac{\sqrt{15}}{2})^2}\\)
=\\frac{15\sqrt{15}}{8}\\) + \\frac{15}{4}\\)
=\\frac{15{(\sqrt{15}+2)}}{8}\\)