How do you simplify #(2x + 3)/( x + 3) + x/ (x - 2)#?

Question

Kennedy Adams · Accepted Answer

It is a little strange:

#(1/3)((3x+(1+sqrt(19)))(3x+(1-sqrt(19))))/((x+3)(x-2))# #(2x+3)/(x+3)+x/(x-2)# #=(x-2)/(x-2)(2x+3)/(x+3)+(x+3)/(x+3)x/(x-2)# #=(2x^2+3x-4x-6+x^2+3x)/((x+3)(x-2))# #=(3x^2+2x-6)/((x+3)(x-2))# Now we must find the roots of: the denominator: #x=(-2+-sqrt(2^2-4*3*(-6)))/(2*3)# #x=(-2+-sqrt(76))/(6)# #x=(-1+-sqrt(19))/(3)# So the polynomium simplifies to: #3((x+(1+sqrt(19))/(3))(x+(1-sqrt(19))/(3)))/((x+3)(x-2))# #(1/3)((3x+(1+sqrt(19)))(3x+(1-sqrt(19))))/((x+3)(x-2))#

Eva Abramson · Answer

undefined To simplify the expression (2x + 3)/(x + 3) + x/(x - 2), we need to find a common denominator for the two fractions. The common denominator is (x + 3)(x - 2).

To do this, we multiply the first fraction by (x - 2)/(x - 2) and the second fraction by (x + 3)/(x + 3).

This gives us ((2x + 3)(x - 2))/((x + 3)(x - 2)) + (x(x + 3))/((x + 3)(x - 2)).

Next, we can combine the numerators over the common denominator:

((2x + 3)(x - 2) + x(x + 3))/((x + 3)(x - 2)).

Expanding and simplifying the numerator gives us (2x^2 - x - 6 + x^2 + 3x)/((x + 3)(x - 2)).

Combining like terms in the numerator gives us (3x^2 + 2x - 6)/((x + 3)(x - 2)).

Therefore, the simplified expression is (3x^2 + 2x - 6)/((x + 3)(x - 2)).