Solve the equation after making an appropriate substitution. left parenthesis 11 x minus 1 right parenthesis squared minus 2 left parenthesis 11 x minus 1 right

Question

Solve the equation after making an appropriate substitution.
left parenthesis 11 x minus 1 right parenthesis squared minus 2 left parenthesis 11 x minus 1 right parenthesis minus 3 equals 0(11x−1)2−2(11x−1)−3=0

Answer 1

I think I understand the problem to be (11x - 1)² - 2(11x - 1) - 3 = 0

 

Let u = (11x - 1)

 

We can rewrite this as u² - 2u - 3 = 0

 

This factors as (u + 1)(u - 3) = 0

 

u + 1 = 0

u = -1

 

and u - 3 = 0

u = 3

 

Now, we know u = 11x - 1

11x - 1 = -1

11x = 0

x = 0

 

and 

 

11x - 1 = 3

11x = 4

x = 4/11

 

The two solutions to this equation are x = 0 and x = 4/11

Answer 2

(11x - 1)² - 2(11x - 1) - 3 = 0

 

 

Notice that this is a quadratic equation.  Without performing any expansion of terms, we can use substitution.

 

Let    q=11x-1

 

Then,

 

q² - 2q - 3 = 0

 

 

Solve for q from this equation.

 

Factor using FOIL.

 

(q - 3)(q + 1) = 0

 

 

Setting the factors equal to zero gives us.

 

q = 3         and        q = -1

 

 

Now substitute these values to solve for x.

 

11x - 1 = -1             and             11x - 1 = 3

 

 

Take it from here.