Algebra and Quadratics?

Question

Hello,

There are some questions that I am not sure how to do:

1) Rationalise the denominators:

1a) 4 / (1 - √3)

1b) (√5 + 2) / (√7 - 3)

1c) (3√5 + 2) / (2√5 - 4)

2) Factorise: 6 + x - 2x^2 (Please explain this)

3) Factorise: 6x^2 - 7x - 3 = 0 (Please explain this)

Thank you in advance.

Answer 1

1a)

Rationallization  of 4 / (1 - √3)

    

Multiply numerator and  denominator with (1 +√3)

1b)

Rationallization of (√5 + 2) / (√7 - 3)

Multiplying numerator and  denominator  with (√7 +3)

 

2)

Factorization of : 6 + x - 2x² = 0

Rewrite the equation as 2x² - x - 6 = 0

2x² - 4x+ 3x- 6=0

2x(x-2) + 3(x-2)=0

(x-2)(2x+3)=0

2x+3=0 ,x - 2=0

Ans x=2, x= -3/2

3)

Factorization of : 6x² - 7x - 3 = 0

6x² - 9x +2x-3=0. 

3x( 2x-3) +1(2x-3)=0. 

(3x+1) (2x-3) =0. 

3x+1=0

x= -1/3.

2x-3=0

x= 3/2.

Ans  x= -1/3, 3/2