Question

Which quadratic equals is equivalent to (x-4)^2 -(x-4)-6=0

Answer 1

t = (x - 4) → t² - t - 6 = 0

t = 3 → x = 7

Explanation:

`(x-4)^2-(x-4)-6=0\\\\\text{substitute:}\ t=(x-4)\geq0\\\\t^2-t-6=0\\\\t^2+2t-3t-6=0\\\\t(t+2)-3(t+2)=0\\\\(t+2)(t-3)=0\iff t+2=0\ \vee\ t-3=0\\\\t+2=0\qquad\text{subtract 2 from both sides}\\\\t=-2<0\to\text{It's not a solution}\\\\t-3=0\qquad\text{add 3 to both sides}\\\\\boxed{t=3}>0\to\text{It's the solution}`

`\text{We're going back to substitution:}\\\\t=3\to x-4=3\qquad\text{add 4 to both sides}\\\\\boxed{x=7}`