Question Number 104588 by  M±th+et+s last updated on 22/Jul/20

solve in R  ((96x−24)/(12x+5))=(√(−144x^2 +72x+7))

Commented by M±th+et+s last updated on 22/Jul/20

thanks both for solutions

Answered by ajfour last updated on 22/Jul/20

8−(((64)/(12x+5)))=(√(16−(12x−3)^2 ))  let  12x−3=t   with   −4≤t≤4  ⇒  8−((64)/(t+8))=(√(16−t^2 ))  ⇒  4−(√(16−t^2 ))=((64)/(t+8))−4     ....(i)  ⇒   (t^2 /(4+(√(16−t^2 )))) = ((64)/(t+8))−4  ⇒  4+(√(16−t^2 )) = (t^2 /4)(((t+8)/(8−t)))   ...(ii)  Adding  (i) & (ii)       8=(t^2 /4)(((t+8)/(8−t)))+4(((8−t)/(t+8)))  Now let  ((8−t)/(t+8)) = s  ⇒  t=8(((1−s)/(s+1)))  ⇒   8=16(((1−s)/(s+1)))^2   ((1/s))+4s  ⇒    2s(s+1)^2 =4(1−s)^2 +s^2 (s+1)^2   ⇒   s(s+1)^2 (2−s)=4(1−s)^2   ⇒   (s+1)^2 [1−(s−1)^2 ]=4(1−s)^2   ⇒   (1/((1−s)^2 ))−(4/((1+s)^2 )) = 1  ......