solve. squreroot(x-1)=x-12?

Answer 1

√(x-1) = x-12
x-1 = (x-12)² = x² - 24x + 144
x² - 25x + 145 = 0

Using the quadratic formula:

x = (25± √(625 - 580))/2 = (25±√45)/2 = (25±3√5)/2

Of course, this only shows that IF √(x-1) = x-12, THEN (25±3√5)/2. Since one of the steps we used at the beginning, namely that of squaring both sides, was not invertible, it is possible that the converse implication does not hold. Of course, since all the steps after that ARE invertible, both of these solutions satisfy x-1 = (x-12)², and thus √(x-1) = √((x-12)²), but because (25 - 3√5)/2 < (25 - 6)/2 = 19/2 < 12, we have (25 - 3√5)/2 - 12 < 0, so √((25 - 3√5)/2 - 1) = √(((25 - 3√5)/2 - 12)²) = 12 - (25 - 3√5)/2 ≠ (25 - 3√5)/2 - 12, thus (25 - 3√5)/2 is extraneous. Of course (25 + 3√5)/2 - 12 > 0, so this solution works.

Thus the sole correct solution is (25 + 3√5)/2. And we are done.

Answer 2

square both sides:
x-1 = x^2 -24x + 144

0 = x^2 - 25x +145

use the quadratic equation

x = (-b sqrt(b^2 +/- 4ac))/2a