Find two number that total 12 and whose squares total 90

Question

use factoring to solve

Answer 1

Let the two numbers be x and y.

 

"two number that total 12"      means       x + y = 12        [eq1]

 

"whose squares total 90"      means       x² + y² = 90       [eq2]

 

The math:

 

Use eq1 to isolate one variable:

     y = 12 - x

 

Now, substitute for y in eq2:

    x ²  +  (12 - x)² = 90

    x²   +  144 - 24x + x² = 90             [use F-O-I-L]

   2x²  - 24x + 54  =  0                  [collect terms]

    x²  - 12x  +  27  =  0                 [divide by 2]

   (x - 3)(x - 9) = 0                    [factor]

Either (x-3)=0 or (x-9)=0              [Multiplicative Property of Zero]

    Either x=3 or x=9

 

Check: 

        Is   3+9=12   ?

              12 = 12   ?yes

        Is    3² + 9² = 90   ?

                9 + 81 = 90   ?

                   90 = 90   ?yes

Answer 2

You have two equations:

 

x + y = 12

 

x²  + y² = 90

 

substitute for y in the second equation:

 

x² + (12-x)² = 90

x²+ 144 + x²-24x = 90

2x²-24x+54=0

x²-12x+27=0

re-arranging so that we can factorize the above equation:

x²-9x-3x+27=0

factorizing

x(x-9)-3(x-9)=0

(x-9)(x-3)=0

 

So, one number is 3 and the other is 9

 

 

 

 

 

Answer 3

9^{2 = 81}

3^{2 = 9}

9+3=12

81+8=90