Rewrite the following expression in the form a Squareroot b

Rewrite the following expression in the form a Squareroot b where b and b are integers: Squareroot75 + Squareroot48. Rewrite the following expression so that each variable appears only once, so that no radicals appear, and: there are no negative exponents. Completely factor the following polynomials -9x^3 - 3x^2 + 3x + 1 3x^3 - 27x Write each of the following as a single, non compound rational expression in lowest terms. You rr numerators and denominators in standard form or in completely factored form. Write down the general solution of ax^2 + bx + c = 0. Then, use it to solve the following equation: x^2 + 1 = 4x. Solve the following equations and inequalities over the real numbers, using algebra. Write solution se interval notation when appropriate.

Solution

4) a )

(x+3) / ( 4x^2 - 9 ) divide by (x^2+7x+12) / ( 2x^2 - 7x + 3)

factoring each polynomial

4x^2 - 9 = (2x-3) (2x+3)

x^2 + 7x + 12 = (x+4)(x+3)

2x^2 -7x + 3 = (2x-1)(x-3)

hence the rational expression now becomes

(x+3) / (2x-3) (2x+3) divide by (x+4)(x+3) / (2x-1)(x-3)

(x+3) / (2x-3) (2x+3) * (2x-1)(x-3) / (x+4)(x+3)

(2x-1)(x-3) / ( 2x-3)(2x+3)(x+4)

b) { x / (x^2 + x- 2) } - {2 / ( x^2 - 5x + 4) }

factors of x^2 + x - 2 = (x+2)(x-1)

factors of x^2-5x+4 = (x-1)(x-4)

therefore lcd is (x+2)(x-1) (x-4)

{x (x-4) - 2 ( x+2) } / (x+2)(x-1) (x-4)

x^2 - 4x -2x - 4 / (x+2)(x-1) (x-4)

(x^2 -6x - 4 ) / (x+2)(x-1) (x-4)

5) general solution of ax^2 + bx + c = 0

x = {-b + - sqrt ( b^2 - 4ac ) } / 2a

x^2 + 1 = 4x

subtracting 4x from both sides

x^2 - 4x + 1 = 0

a = 1 , b = -4 , c = 1

plugging the values of a,b,c in the quadratic formula

x = {- ( -4) + - sqrt ( (-4)^2 - 4* 1* 1 }/ 2* 1

x = { 4 + - sqrt 12 } / 2

x1 = 2+ sqrt 3

x2 = 2 - sqrt 3