Solution To Equation Involving Square Root: Extraneous Solution?

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SUMMARY

The solution set for the equation sqrt{x+2} = x - 4 is definitively x = 7. The value x = 2 is an extraneous solution introduced by squaring both sides of the equation. Verification shows that when x = 7, both sides equal 3, confirming its validity, while x = 2 results in a mismatch, as the left side equals 2 and the right side equals -2. Therefore, only x = 7 satisfies the original equation.

PREREQUISITES
  • Understanding of square root functions and their properties
  • Knowledge of solving equations involving squaring both sides
  • Familiarity with the concept of extraneous solutions in algebra
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of square root functions in detail
  • Learn about identifying and eliminating extraneous solutions in equations
  • Explore quadratic equations and their solutions, including the use of the quadratic formula
  • Practice solving similar equations involving square roots and verify solutions
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Students learning algebra, educators teaching mathematical concepts, and anyone interested in mastering equation solving techniques.

rebo1984
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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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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.
 

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