Answer:
There is only one solution: x=10
The solution x=4 is an extraneous solution.
Step-by-step explanation:
We have this equation:
[tex]\sqrt{2x-4} -x+6=0[/tex]
Let's solve for x:
Add x to both sides:
[tex]\sqrt{2x-4} -x+x+6=0+x\\\sqrt{2x-4}+6=x[/tex]
Subtract 6 from both sides:
[tex]\sqrt{2x-4}+6-6=x-6\\\sqrt{2x-4}=x-6[/tex]
Raise both sides to the power of two:
[tex]2x-4=(x-6)^2[/tex]
Expand out the terms of the right side:
[tex]2x-4=x^2-12x+36[/tex]
Subtract 2x from both sides and add 4 to both sides:
[tex]x^2-14x+40=0[/tex]
Factor into a product:
[tex](x-10)(x-4)=0[/tex]
So the solutions are:
[tex]x_1=10\\x_2=4[/tex]
Evaluating [tex]x_1[/tex] :
[tex]\sqrt{2*10-4} -10+6=0\\4-10+6=0\\0=0[/tex]
This solution is correct.
Evaluating [tex]x_2[/tex] :
[tex]\sqrt{2*4-4} -4+6=0\\2-4+6=0\\4=0[/tex]
So this solution is incorrect.
Therefore:
There is only one solution: x=10
The solution x=4 is an extraneous solution.
