Extra Questions For Class 9 Math Chapter 1 | Q9

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

Video Explanation

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})