First, multiply each side of the equation by #color(red)(12)color(red)(x)# to eliminate the fractions while keeping the equation balanced:
#color(red)(12)color(red)(x)(37/12 + 1/4x) = color(red)(12)color(red)(x) xx 1/(6x)#
#(color(red)(12)color(red)(x) xx 37/12) + (color(red)(12)color(red)(x) xx
1/4x) = cancel(color(red)(12))cancel(color(red)(x))2 xx 1/(color(red)(cancel(color(black)(6)))color(red)(cancel(color(black)(x))))#
#(cancel(color(red)(12))color(red)(x) xx 37/color(red)(cancel(color(black)(12)))) + (cancel(color(red)(12))3color(red)(x) xx
1/color(red)(cancel(color(black)(4)))x) = 2#
#37x + 3x^2 = 2#
Now, subtract #color(red)(2)# from each side of the equation and put the equation in standard form while keeping the equation balanced:
#37x + 3x^2 - color(red)(2) = 2 - color(red)(2)#
#37x + 3x^2 - 2 = 0#
#3x^2 + 37x - 2 = 0#
We can now use the quadratic equation to solve this problem:
The quadratic formula states:
For #color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0#, the values of #x# which are the solutions to the equation are given by:
#x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))#
Substituting:
#color(red)(3)# for #color(red)(a)#
#color(blue)(37)# for #color(blue)(b)#
#color(green)(-2)# for #color(green)(c)# gives:
#x = (-color(blue)(37) +- sqrt(color(blue)(37)^2 - (4 * color(red)(3) * color(green)(-2))))/(2 * color(red)(3))#
#x = (-color(blue)(37) +- sqrt(1369 - (-24)))/6#
#x = (-color(blue)(37) +- sqrt(1369 + 24))/6#
#x = (-color(blue)(37) +- sqrt(1393))/6#
