Question

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

Answer 1

`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`

Answer 2

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}`

Answer 3

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]