First, put the equation in standard form:
#30x^3 - color(red)(21x^2) - color(blue)(135x) = 21x^2 - color(red)(21x^2) + 135x - color(blue)(135x)#
#30x^3 - 21x^2 - 135x = 0 + 0#
#30x^3 - 21x^2 - 135x = 0#
Next, divide each side of the equation by #color(red)(3)# to minimize the coefficients:
#(30x^3 - 21x^2 - 135x)/color(red)(3) = 0/color(red)(3)#
#(30x^3)/color(red)(3) - (21x^2)/color(red)(3) - (135x)/color(red)(3) = 0#
#10x^3 - 7x^2 - 45x = 0#
Then, factor out the common term:
#(x * 10x^2) - (x * 7x) - (x * 45) = 0#
#x(10x^2 - 7x - 45) = 0#
Now, solve each term on the left for #0# to find the solutions:
Solution 1:
#x = 0#
Solution 2:
We can use the quadratic equation to solve for this term:
#color(red)(10)x^2 - color(blue)(7)x - color(green)(45) = 0#
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)(10)# for #color(red)(a)#
#color(blue)(-7)# for #color(blue)(b)#
#color(green)(-45)# for #color(green)(c)# gives:
#x = (-color(blue)(-7) +- sqrt(color(blue)(-7)^2 - (4 * color(red)(10) * color(green)(-45))))/(2 * color(red)(10))#
#x = (7 +- sqrt(49 - (-1800)))/20#
#x = (7 +- sqrt(49 + 1800))/20#
#x = (7 +- sqrt(1849))/20#
#x = (7 +- 43)/20#
#x = (7 - 43)/20#; #x = (7 + 43)/20#
#x = -36/20#; #x = 50/20#
#x = -9/5#; #x = 5/2#
The Solution Set Is:
#x = {-9/5, 0, 5/2}#