the product of some negative number and 2 less than three times that number is 21. Find the number.

Question

 Type as an integer or a fraction?

Answer 1

It is important to correctly translate word (story) problems into concise, precise math expressions (with variables and operations).  Note that English is often ambiguous.

 

product       means      *

less than                     -

times                          *

is                                =

.. (add to this list with each problem you do)

 

Let x= the number

 

Translate

  "the product of some negative number and 2 less than three times that number is 21"

   the product of                x                     and 2 less than three times that number is 21

   the product of                x                    and 2 less than           3*x                          is 21

   the product of                x                    and 2 less than           3*x                         = 21

   the product of              x                    and     ( -2   +    3*x)                                 = 21

                                       x                     *        ( -2 + 3*x)                                      = 21

rewrite:

             x(3x-2) = 21

 

             3x² - 2x - 21 = 0             [distribute; collect terms]

            (3x + 7)(x - 3) = 0           [factor or quadratic formula]

   either x=-7/3   or  x=3

 

       x = -7/3                ["some negative number"]

 

Answer 2

x(3x-2) = 21       x<0

 

3x² - 2x = 21

3x² - 2x - 21 = 0

 

We can factor by grouping.

3(-21) = -63

Factors of -63 that add to -2 are (-9)(7)

Replace -2x with -9x + 7x

 

3x²-9x+7x-21 = 0

3x(x-3) + 7(x-3) = 0

(3x+7)(x-3) = 0

 

Either 3x+7=0

          3x = -7

          x = -7/3

 

OR   x-3 = 0

       x = 3

 

Since we are told the number must be negative then:

 

x = -7/3

   

Answer 3

Let n be the number.

n(3n-2)=21
3n²-2n=21
3n²-2n-21=0
Using the quadratic formula:
n = (-b±√(b²-4ac))/2a
where a=3, b=2, c=-21
n = (-2±√((-2)²-(4*3*-21)))/(2*3)

n = +3,-7/3

Since the question is asking for negative number, ignore +3.

The number is -7/3