MHB Solution To Equation Involving Square Root: Extraneous Solution?

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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
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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.
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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