How do you solve by factoring #30x^3 = 21x^2 + 135x#?

3 Answers
Jul 8, 2018

Answer:

#x_1=-5/9#, #x_2=0# and #x_3=5/2#

Explanation:

#30x^3=21x^2+135x#

#30x^3-21x^2-135x=0#

#3x*(10x^2-7x-45)=0#

#3x*(10x^2-25x+18x-45)=0#

#3x*[5x*(2x-5)+9*(2x-5)]=0#

#3x*(5x+9)*(2x-5)=0#

So, #x_1=-5/9#, #x_2=0# and #x_3=5/2#

Jul 8, 2018

Answer:

See a solution process below:

Explanation:

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}#

Jul 8, 2018

Answer:

Solution: # x=0 , x=2.5 , x= -1.8#

Explanation:

#30 x^3=21 x^2+135 x# or

#30 x^3 -21 x^2-135 x = 0# or

#3 x(10 x^2 -7 x-45 ) = 0# or

#3 x(10 x^2 -25 x +18 x-45 ) = 0# or

#3 x{5 x(2 x -5) +9(2 x-5) } = 0# or

# x(2 x -5)(5 x+9) = 0#

# :. x=0 , 2 x-5=0 :. x= 5/2=2.5 and 5 x+9=0 #

#:. x = -9/5 = -1.8 #

Solution: # x=0 , x=2.5 , x= -1.8# [Ans]