Question

(x-5)(x-6)=25/576 solve for x

Answer 1

(x-5)(x-6) = 25/567
⇒  x²-6x-5x+30 = 5/24
x²-10x+25-(x-5) = 5/24 
⇒  (x-5)²-(x-5)-(5/24)² = 0
⇒   y²-y-(5/24)² = 0
⇒   y²-25y/24+y/24-5²/24² = 0
⇒      y(y-25/24) 1/24(y-25/24) = 0
⇒      (y-25/24)(y+1/24) = 0
⇒     y-25/24 = 0 ,   then  y = 25/24
therefore,    x-5 = 25/24 ,⇒  x = 25/24+5 = 145/24
    And  y+1/24 = 0,   then y = -1/24
therefore,   x-5 = -1/24 ,  ⇒   x = 5-1/24 = 119/24
the values of x are 145/24 and 119/24

Answer 2

Take (x-6) = t
 then (x-5) = (t+1)

(x-5)+(x-6) = 25/576
⇒(t+1)t = 25/576

⇒t² + t - $\frac{25}{576}$ = 0

t = $\frac{-1+ \sqrt{1^2-4*1*(-25/576)} }{2*1}$ and $\frac{-1+ \sqrt{1^2-4*1*(-25/576)} }{2*1}$

or t = $\frac{1}{24}$ and $\frac{-25}{24}$

But t = (x-6)
case: 1 
t = 1/24
⇒x-6 = 1/24
⇒x = 6 + 1/24
⇒x = (144+1)/24
⇒x = 145/24
case: 2
t = -25/24
⇒x-6 = -25/24
⇒x = 6 -25/24
⇒x = (144-25)/24
⇒x = 119/24