Question

Solve (x + 1)2 - 4(x + 1) + 2 = 0 using substitution.

Answer 1

Answer:

Step-by-step explanation:

We are given the equation $(x+1)^2-4(x+1)+2=0$. Consider the substitution u = (x+1). Then the equation turns out to be $u^2-4u+2=0$

Recall that given a second degree polynomial of the form $ax^2+bx+c =0$ then its solutions are given by the expression

$x = \frac{-b \pm \sqrt[]{b^2-4ac}}{2a}$

In our case, a = 1, b = -4 and c =2. Then the solutions are

$u_1 = \frac{4+\sqrt[]{(-4)^2-4(2)}}{2} = 2 +\sqrt[]{2}$

$u_2 = \frac{4-\sqrt[]{(-4)^2-4(2)}}{2} = 2 -\sqrt[]{2}$

We have that x = u-1. So the original solutions are

$x_1 = 2 +\sqrt[]{2}-1 = 1 + \sqrt[]{2}$

$x_2 =2 -\sqrt[]{2} -1 = 1 - \sqrt[]{2}$