MHB Solution To Equation Involving Square Root: Extraneous Solution?

Click For Summary
The equation sqrt{x+2}= x-4 has a solution set of only x=7, as x=2 is an extraneous solution. When substituting x=7, both sides of the equation equal 3, confirming it as a valid solution. In contrast, substituting x=2 results in a mismatch, as the left side equals 2 while the right side equals -2. The discussion emphasizes that the square root function only yields non-negative results, which is crucial in solving such equations. Therefore, the correct solution is x=7.
rebo1984
Messages
18
Reaction score
0
Hi everyone,

What is the solution set of the equation: sqrt{x+2}= x-4

I got 2 and 7.

Is it correct or is it just 7. If so why?

Thanks:)
 
Mathematics news on Phys.org
Re: Solution to equations

It's just seven. You're squaring the RHS which introduces an extraneous solution.
 
Re: Solution to equations

rebo1984 said:
Hi everyone,

What is the solution set of the equation: sqrt{x+2}= x-4

I got 2 and 7.

Is it correct or is it just 7. If so why?

Thanks:)
Check:
if x= 7 then sqrt(x+ 2)= sqrt(7+ 2)= sqrt(9)= 3 while x- 4= 7- 4= 3. Those are the same so x= 7 satisfies sqrt(x+ 2)= x- 4.

If x= 2, sqrt(x+ 2)= sqrt(2+ 2)= sqrt(4)= 2 while x- 4= 2- 4= -2. Those are not the same so x= 2 does not satisfy sqrt(x+ 2)= x- 4.

Note that the square root function, sqrt(4), cannot give both "2" and "-2" because a function, by definition, must give a single value. Further, suppose you were asked to solve the equation x^2= a. There are two numbers that satisfy that equation which you would write as x= +/- sqrt(a). The reason you need the "+/-" is because sqrt(a) is only the positive solution.
 
Re: Solution to equations

Thank you for your very detailed response.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

High School Complete the square, yes, but what does it mean?
  • · Replies 19 ·
Replies
19
Views
3K
High School Principal square root of a complex number, why is it unique?
  • · Replies 13 ·
Replies
13
Views
6K
Undergrad Trouble Solving an Equation that has square roots on both sides
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Integer Square Roots of P{0,1,2,4,5,6,7,8,9}
  • · Replies 23 ·
Replies
23
Views
2K
High School Determine a fractional square root without calculator
  • · Replies 15 ·
Replies
15
Views
2K
High School Problem with simple inequality
  • · Replies 13 ·
Replies
13
Views
2K
High School Why Must the Expression Inside a Square Root Be Non-Negative?
  • · Replies 16 ·
Replies
16
Views
2K
MHB Is this theory regarding the graph and the square root valid?
  • · Replies 1 ·
Replies
1
Views
2K
High School What went wrong with (-x)^2=x^2?
  • · Replies 22 ·
Replies
22
Views
2K
MHB Radical Equation...Extraneous Roots
  • · Replies 5 ·
Replies
5
Views
2K