The sum of a number and its square is 12. Find its number

Question

The sum of a number and its square is 12. Find its number

Answer 1

Let x = the unknown number.  Its square is x².  The sum of a number and its square is 12:

 

     x + x² = 12

 

Putting the quadratic into standard form:

 

     x² + x - 12 = 0

     (x+4)(x-3) = 0

     x = -4 and 3

 

Let's check x=-4:

     x + x² = 12

     (-4) + (-4)² = 12

     -4 + 16 = 12

     12 = 12

 

Now let's check x= 3:

     x + x² = 12

     3 + 3² = 12

     3 + 9 = 12

     12 = 12

 

Yup, they both work. There are two answers, x = -4 and x = 3

 

Answer 2

 

Let n = the number

The square of the is n²

The sum of the number and its square is n² + n

 

   n² + n = 12

   n² + n - 12 = 0

 

Now factor and solve using the zero product property

 

   (n - 3)(n+ 4) = 0

   

   n - 3 = 0 ...  n = 3

 

   n + 4 = 0 ... n = -4

 

CHECK:

   (3) :  3 + 3 =3 + 9 = 12

  (-4):  -4 + (-4)² = -4 + 16 = 12