how do you solve: 6x cubed -27x squared - 105x = 0?

x(6x² - 27x - 105) = 0
x(2x² - 9x - 35) = 0
x[2x + 5)(x - 7)] = 0

x = 0, -5/2 or 7

6x³ - 27x² - 105x = 0
3x(2x² - 9x - 35) = 0
x² - 9/2x = 35/2
x² - 9/4x = 35/2 + (- 9/4)²
x² - 9/4x = 280/16 + 81/16
(x - 9/4)² = 361/16
x - 9/4 = 19/4

Values of x:
x = 19/4 + 9/4, x = 28/4, x = 7
x = - 19/4 + 9/4, x = - 10/4, x = - 5/2
3x = 0, x = 0

Answer: x = 7, - 5/2, 0

Proof (x = 7):
6(7³) - 27(7²) = 105(7)
6(343) - 27(49) = 735
2,058 - 1,323 = 735
735 = 735

Proof (x = - 5/2):
6(- 5/2)³ - 27(- 5/2)² = 105(- 5/2)
6(- 15.625) - 27(6.25) = - 262.5
- 93.75 - 168.75 = - 262.5
- 262.5 = - 262.5

Proof (x = 0):
6(0³) - 27(0²) = 105(0)
0 - 0 = 0

6x^3 - 27x^2 - 105x =

Factor each term:
6x^3 = 2 * 3 * x * x^2
-27x^2 = -9 * 3 * x * x
-105x = -35 * 3 * x

Notice that 3 * x = 3x is a factor for each of them. Factor that out.
3x(2x^2 - 9x - 35) = 0

Replace (2x^2 - 9x - 35) with 2 sets of parentheses.
3x(_________)(_________) = 0

Look at the first term's factors.
2x^2 = 2x * x = 2x^2 * 1

Since the middle term is an x term, you want the first pair. Put 2x and x at the beginning of the parentheses sets.
3x(2x_____)(x_____) = 0

Look at the last term's factors. You want the pair that will add up to the middle term -9x when multiplied by the 2x and x you already have.
1 * -35 ===> (2x * 1) + (1x * -35) = 2x + -35x = -33x <== wrong pair
5 * -7 ===> (2x * 5) + (1x * -7) = 10x + -7x = 3x <== wrong pair
7 * -5 ===> (2x * 7) + (1x * -5) = 14x + -5x = 9x <== wrong pair
35 * -1 ===> (2x * 35) + (1x * -1) = 70x + -1x = 69x <== wrong pair
-1 * 35 ===> (2x * -1) + (1x * 35) = -2x + 35x = 33x <== wrong pair
-5 * 7 ===> (2x * -5) + (1x * 7) = -10x + 7x = -3x <== wrong pair
-7 * 5 ===> (2x * -7) + (1x * 5) = -14x + 5x = -9x <== right pair
-35 * 1 ===> (2x * -35) + (1x * 1) = -70x + 1x = -69x <== wrong pair

The right pair is -7 * 5.

Since 2x is paired with -7 and the 2x is on the outside, write -7 on the outside.
3x(2x_____)(x - 7) = 0

Since the x is paired with the 5 and the x is on the inside, write +5 on the inside.
3x(2x + 5)(x - 7) = 0

Solve.
3x = 0
x = 0 / 3 = 0

2x + 5 = 0
2x = -5
x = -5 / 2

x - 7 = 0
x = 7

ANSWER: x = 0, (-5 / 2), or 7

6x^3 - 27x^2 - 105x = 0 ...eq. [1]

Factor out a '3x' from each term on the left-hand side of eq. [1]:

3x[2x^2 - 9x - 35] = 0 ...eq. [2]

Eq. [2] is satisfied when either term - (3x) or (2x^2 - 9x - 35) - is equal to zero. So, set each term equal to zero and solve:

3x = 0
x = 0 ...[sol'n 1]

2x^2 - 9x - 35 = 0 ...eq. [3]

I probably wouldn't bother trying to factor eq. [3] unless you see something right away. I would just use the Quadratic Formual with a = 2, b = -9, and c = -35. Doing this results in the following solutions:

x = [9 + sqrt(361)] / 4 = 7 ...[sol'n 2]
x = [9 - sqrt(361)] / 4 = -5 / 2 ...[sol'n 3]

So, the solutions are:

x = -5 / 2, 0, and 7

Hope this helps!

6x^3 - 27x^2 - 105x = 0
x(6x^2 - 27x - 105) = 0
3x(2x^2 - 9x - 35) = 0
3x(2x^2 + 5x - 14x - 35) = 0
3x[(2x^2 + 5x) - (14x + 35)] = 0
3x[x(2x + 5) - 7(2x + 5)] = 0
3x(2x + 5)(x - 7) = 0

3x = 0
x = 0/3
x = 0

2x + 5 = 0
2x = -5
x = -5/2 (-2.5)

x - 7 = 0
x = 7

∴ x = -5/2 (-2.5) , 0 , 7

6x^3 - 27x^2 -105x = 0

3x(2x^2 - 9x - 35) = 0
3x(2x + 5)(x - 7) = 0

x = 0, -2/5, 7