Extra Questions For Class 9 Math Chapter 1 | Q9

Number Systems | Rationalize Irrational Number & Cubic Expression

Question 9: If x = \\frac{1}{8-\sqrt{60}}), what is the value of x3 - 5x2 + 8x - 4?


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Explanatory Answer | Number Systems

In all these questions, before plugging the value of x in the expression that has to be evaluated, rationalize x and get x expressed as a number with a rational denominator.
You are likely to find a clue after completing this step that will make things easier to solve.

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 x3 - 5x2 + 8x - 4

Because x = 2 + \\frac{\sqrt{15}}{2}), try and express (x3 - 5x2 + 8x - 4) in terms of (x - 2)3, (x - 2)2 and (x - 2).
(x - 2)3 = x3 - 6x2 + 12x - 8 -------- (1)
(x - 2)2 = x2 - 4x + 4 -------- (2)

Add or subtract the two equations to check whether you get the given expression
(1) + (2) = x3 - 5x2 + 8x - 4
So, x3 - 5x2 + 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})

 


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