Answer :
Given the equation:
[tex]x-1=\sqrt[]{4x-4}[/tex]Let's solve the equation for x.
To solve for x, take the following steps:
• Step 1.
Square both sides:
[tex]\begin{gathered} (x-1)^2=\sqrt[]{4x-4}^2 \\ \\ (x-1)(x-1)=4x-4 \\ \end{gathered}[/tex]• Step 2.
Expand using FOIL method and distributive property
[tex]\begin{gathered} x(x-1)-1(x-1)=4x-4 \\ \\ x(x)+x(-1)-1(x)-1(-1)=4x-4 \\ \\ x^2-x-x+1=4x-4 \\ \\ x^2-2x+1=4x-4 \end{gathered}[/tex]• Step 3.
Add 4 to both sides and also subtract 4x from both sides:
[tex]\begin{gathered} x^2-2x-4x+1+4=4x-4x-4+4 \\ \\ x^2-6x+5=0 \end{gathered}[/tex]• Step 4.
Factor the left side of the equation using the AC method.
Find two numbers whose sum is -6 and whose product is 5.
Thus, we have:
-5 - 1 = -6
-5 x -1 = 5
Therefore, the numbers are:
-5, and -1
Hence, we have:
[tex](x-5)(x-1)=0[/tex]Equate trhe individual factors to zero and solve for x:
[tex]\begin{gathered} x-5=0 \\ \text{Add 5 to both sides}\colon \\ x-5+5=0+5 \\ x=5 \\ \\ \\ x-1=0 \\ \text{Add 1 to both sides:} \\ x-1+1=0+1 \\ x=1 \end{gathered}[/tex]Therefore, the solutions to the given equation is:
[tex]x=5,\text{ and 1}[/tex]ANSWER:
x = 5, 1
