Question

If X = 1/2-√3
Then what is the value of x^3 - x^2 - 11x +3

Answer 1

★ Solution :-

Given ,

• x = 1/2 - √3

We need to find ,

• x³ - x² - 11x + 3

Firstly factoring the given expression

→ x³ - x² - 11x + 3

→ x³ + 3x² - 4x² - 12x + x + 3

→ x²(x + 3) - 4x(x + 3) + 1(x + 3)

Taking common

→ (x + 3) ( x² - 4x + 1 )

Now , taking the second term

• x² - 4x + 1

Now , substituting the value of x

➵ ( 1/2 - √3 )² - 4( 1/2 - √3 ) + 1

➵ 1/(2 - √3)² - 4/2 - √3 + 1

➵ [ 1/2² + (√3)² - 2(2)(√3) ] - [ 4/2 - √3 ] + 1

➵ 1/(4 + 3 - 4√3) - [ 4/2 - √3 ] + 1

➵ [ 1/7 - 4√3 ] - [ 4/2 - √3 ] + 1

➵ [ ( 2 - √3 ) - 4( 7 - 4√3 )/(7 - 4√3)(2 - √3) ] + 1

➵ [ 2 - √3 - 24 - 16√3 ]/[ 14 - 7√3 - 8√3 - 12 ] + 1

➵ [ - 22 - 17√3 ]/[ 2 - 15√3 ] + 1

➵ [ - 22 - 17√3 ] + [ 2 - 15√3 ]/[ 2 - 15√3 ]

➵ [ - 20 - 32√3 ]/[ 2 - 15√3 ]

Now , taking first term .

➵ 1/(2 - √3) + 3

➵ 1 + (2 - √3)3/(2 - √3)

➵ 1 + 6 - 3√3/(2 - √3)

➵ [ 7 - 3√3 ]/[ 2 - √3 ]

Now , multiplying both the terms

➮ { [ - 20 - 32√3 ]/[ 2 - 15√3 ] } { [ 7 - √3 ]/[2 - √3] }

➮ [ - 20 - 32√3 ][7 - √3]/[2 - 15√3][2 - √3]

➮ [ - 140 + 20√3 - 224√3 + 288 ]/[ 4 - 2√3 - 30√3 + 45 ]

➮ [ 148 - 202√3 ]/[ 49 - 32√3 ]

Hence , x³ - x² - 11x + 3 = [ 148 - 202√3 ]/[ 49 - 32√3 ] .