SOLUTION: What is the extraneous solution of sqrt(x-3)= (x-5) also sqrt of(x+4)=sqrt(3x)

Question 717301: What is the extraneous solution of sqrt(x-3)= (x-5)
also sqrt of(x+4)=sqrt(3x)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
First we find solutions. Then we determine which, if any, are extraneous.

Square both sides:

Squaring the left side is as easy as it looks. Squaring an expression like the right side is where a lot of mistakes are made. It is not !! To square a two-term expression like (x-5) correctly we must use FOIL on (x-5)(x-5) or use the pattern. I prefer using the pattern:

which simplifies to:

Now that x is out of the square root we can solve for it. This is a quadratic equation so we want one side to be zero. Subtracting x and adding 3 we get:

Next we factor (or use the Quadratic Formula). This factors fairly easily:

From the Zero Product Property:
x-7 = 0 or x-4 = 0
Solving these we get:
x = 7 or x = 4

Now we check. Use the original equation:

Checking x = 7:

Check!
Checking x = 4:

Check fails!
So x = 7 is actually a solution and x = 4 is extraneous.

Square both sides:

x+4 = 3x
Subtract x:
4 = 2x
Divide by 2:
2 = x
Check.

Checking x = 2:

Check!
x = 2 is the solution to this equation. There are no extraneous solutions.

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