Solving systems of equations algebraically

Question

Solve the system algebraically and verify:

 

y=1/2(x-3)²+4

 

Y=-1/2(x-1)²+6

 

Answer 1

(1/2)(x-3)^2 +4 = (-1/2)(x-1)^2 + 6

 

(x-3)^2 + 8 = (-1)(x-1)^2 + 12 <--- multiplies both sides by 2 to clear away the fractions

 

x^2 - 6x + 9 + 8 = (-1) (x^2 - 2x + 1) + 12 <--- FOIL

 

x^2 - 6x + 17 = (-x^2) + 2x - 1 + 12 <--- CLT left side, distributes -1 right side

 

x^2 - 6x + 17 = 11 + 2x - x^2

 

x^2 + x^2 - 6x - 2x + 17 - 11 = 0  <---- everything left side

 

2x^2 - 8x + 6 = 0

 

x^2 - 4x + 3 = 0

 

(x     -     3)(x  -    1  ) = 0

 

x-3=0 ---> x = 3

x-1 = 0 ---> x = 1