Solution To Equation Involving Square Root: Extraneous Solution?

In summary, the solution set for the equation sqrt(x+2)= x-4 is just x= 7. This is because when you plug in x= 7, both sides of the equation are equal. The other solution, x= 2, does not satisfy the equation. This is because the square root function can only give one value for a given input, and in this case, it is not equal to the other side of the equation. This is also why when solving for x^2= a, we need to use the "+/-" symbol to represent both the positive and negative solutions.
  • #1
rebo1984
18
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:)
 
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  • #2
Re: Solution to equations

It's just seven. You're squaring the RHS which introduces an extraneous solution.
 
  • #3
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.
 
  • #4
Re: Solution to equations

Thank you for your very detailed response.
 

Related to Solution To Equation Involving Square Root: Extraneous Solution?

1. What is an extraneous solution?

An extraneous solution is a solution to an equation that does not satisfy the original equation. This can happen when solving equations involving square roots because taking the square root of a number can introduce an additional solution that does not actually work in the original equation.

2. How can I identify an extraneous solution?

To identify an extraneous solution, you can plug the solution back into the original equation and see if it satisfies the equation. If the solution does not work, then it is extraneous. Another way to identify an extraneous solution is to check if the solution makes the radicand (the number inside the square root symbol) negative, which is not allowed.

3. Why do extraneous solutions occur in equations involving square roots?

Extraneous solutions occur in equations involving square roots because taking the square root of a number can introduce an additional solution that does not actually work in the original equation. This can happen because when we square both sides of an equation to solve for the variable, we are essentially introducing a new equation that may have additional solutions.

4. How can I avoid getting extraneous solutions?

To avoid getting extraneous solutions, you can check your solutions by plugging them back into the original equation. Additionally, you can also simplify the equation before taking the square root, which can help eliminate some potential extraneous solutions.

5. Are extraneous solutions always incorrect?

Yes, extraneous solutions are always incorrect because they do not satisfy the original equation. However, they can still provide useful information in the problem-solving process, such as indicating that there may be an error in the solution or that the equation may have no real solutions.

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