First, multiply each side of the equation by #color(red)(x)# to eliminate the fraction while keeping the equation balanced:
#color(red)(x)(3x + 2/x) = color(red)(x) xx -7#
#(color(red)(x) xx 3x) + (color(red)(x) xx 2/x) = -7x#
#3x^2 + (cancel(color(red)(x)) xx 2/color(red)(cancel(color(black)(x)))) = -7x#
#3x^2 + 2 = -7x#
Next, add #color(red)(7x)# to each side of the equation to put the quadratic in standard form:
#3x^2 + color(red)(7x) + 2 = -7x + color(red)(7x)#
#3x^2 + 7x + 2 = 0#
Then factor the quadratic as:
#(3x + 1)(x + 2) = 0#
Now solve each term on the left side of the equation for #0# to solve for #x#:
Solution 1:
#3x + 1 = 0#
#3x + 1 - color(red)(1) = 0 - color(red)(1)#
#3x + 0 = -1#
#3x = -1#
#(3x)/color(red)(3) = -1/color(red)(3)#
#(color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) = -1/3#
#x = -1/3#
Solution 2:
#x + 2 = 0#
#x + 2 - color(red)(2) = 0 - color(red)(2)#
#x + 0 = -2#
#x = -2#
The Solutions Are: #x = -1/3# and #x = -2#
