precal help Find all real solutions to the system of equatio

precal help

Find all real solutions to the system of equations: (Order your answers from smallest to largest x. y = (x - 4)^2 + 8 (x - 4)^2 + (y - 8)^2 = 12 (x, y) = () (x, y) = () Exactly solve the system of equations: (Order your answers from smallest to largest, first by real part, then by imaginary part. Order your answers from smallest to largest x, then from smallest to largest y.) 16x^2 - 95x + y^2 = -92 x^2 - 5x + y^2 = 43 (x, y) = () (x, y) = ()

Solution

y = (x-4)^2 +8

(x-4)^2 +(y-8)^2 = 12

substituting (x-4)^2 in 2nd equation

y-8 +(y-8)^2 = 12

y^2 +64 - 15y -8 = 12

y^2 -15y +44 =0

y^2 -11y -4y +44 =0

y( y-11) -4(y-11) =0

(y-4)(y-11)=0

y = 4 ; y=11

Now substitute to get values of x: y = (x-4)^2 +8

4 = (x-4)^2 +8

(x-4)^2 = -4 (no real solution)

So, y =11

(x-4)^2 +8 = 11

x-4 = +/-sqrt3

x = 4+sqrt3 , x= 4 -sqrt3

Solution set : x = 4 -sqrt3 ; y=11

x = 4+sqrt3 ; y =11