Question

X^2 - 6 x + 9 = 4 root(x^2 - 6 x + 6)​

Answer 1

Hello,

Explanation:

`x^2-6x+9=4√(x^2-6x+6) \\\\(x-3)^2=4√(x^2-6x+9-3) \\\\(x-3)^2=4√((x-3)^2-3) \\\\Let's\ say\ y=x-3\\\\y^2=4√(y^2-3) \ squaring\\\\y^4=16(y^2-3)\\y^4-16y^2+48=0\\\Delta=16^2-4*48=64=8^2\\y^2=4\ or\ y^2=12\\\\(y-2)(y+2)=0\ or\ (y+2√(3) )(y-2√(3) )=0 \\\\y=2,x=5\ or\\\ y=-2, x=-1\ or\\\ y=-2√(3),x=3-2√(3)\ \\or\ y=2√(3),x=3+2√(3)\\\\`

Verifying:

`1) if\ x=-1\ then\\a)\ x^2-6x+9=1+6+9=16\\b)\ 4√(x^2-6x+6) =4√(1+6+6) =4√(13):\ False\\\\2)if\ x=5\\a)\ x^2-6x+9=25-30+9=4\\b)\ 4√(x^2-6x+6) =4√(25-30+6) =4√(1)=4:\ True\\`

`3) if\ x=3-2√(3)\ x^2=21-12√(3) \\a)\ x^2-6x+9=21-12√(3)-18+12√(3)+9=12\\b)\ 4√(x^2-6x+6) =4\sqrt{21-12√(3)-18+12√(3)+6} =4√(9)=12:\ True\\\\\\4) if\ x=3+2√(3)\ x^2=21+12√(3) \\a)\ x^2-6x+9=21+12√(3)-18-12√(3)+9=12\\b)\ 4√(x^2-6x+6) =4\sqrt{21+12√(3)-18-12√(3)+6} =4√(9)=12:\ True\\`

Sol={5,3-2√3,3+2√3}

Answer 2

Answer

X1= 3-2 root3 x2= 1 x3=5 x4= 3 +2 root 3
First swap the sides then simplify the equation
Collect like terms
Move all the expression to the left to equal 0
Then collect like terms again
Reorder the terms from ^2 to ^4
Factor the expression
Change the signs
Separate into possible cases
Then sold the equation to equal 0
Then check solutions by subbing them in as x
Then you should have 4 perfect answers