Question

If x= 1/2-√3 ,find the value of x³ - 2x² - 7x + 5

Answer 1

x=1/(2-√3)

Then; rationalise x,

x=(2+√3)/[(2-√3)(2+√3)

Or, x= (2+√3)/(4-3)

Or, x=2+√3

Now putting the value of x in the expression we get;

$x^3-2x^2-7x+5$

$=(2+\surd 3{)}^3-2(2+\surd 3{)}^2-7(2+\surd 3)+5$

$=(2+\surd 3)\left[\right(2+\surd 3{)}^2-2(2+\surd 3)-7]+5$

$=(2+\surd 3)[4+3+4\surd 3-4-2\surd 3-7]+5$

$=(2+\surd 3)[-4+2\surd 3]+5$

$=-2(2+\surd 3)(2-\surd 3)+5$

$=-2\left(1\right)+5$

= 3

So the required answer is 3.