What is the sum of the solutions to the above equation?

Algebra

What is the sum of the solutions to the above equation?

Postby Guest » Mon Oct 05, 2020 9:43 am

[tex]\sqrt{4x+20}[/tex]=x+2

I solved this equation by isolating 0 and using the quadratic equation. It came out to x= + or - 4

I plugged in the potential answer to rule out extraneous solutions, and found that both of them could work. However, Khan Academy ruled -4 as an extraneous solution.

When you plug -4 into the equation, you get:

\sqrt{4} = -2

If you square both sides of the equation, this is true. However, if you simplify the square root, the equation is not true. In this case, how do you know how to determine an extraneous solution? It seems if you can square it and it be correct, then it isn't.

Any help is appreciated.
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Re: What is the sum of the solutions to the above equation?

Postby nathi123 » Mon Oct 05, 2020 11:27 am

[tex]\begin{array}{|l} \sqrt{4x+20}=x+2\\ x +2\ge0 \\ 4x+20\ge 0 \end{array}\Leftrightarrow \begin{array}{|l} 4x + 20= x^{2}+4x+4 \\ x\ge -2 \end{array}[/tex]
[tex]\Leftrightarrow \begin{array}{|l} x^{2} -16 = 0\\ x \ge - 2\end{array} \Leftrightarrow \begin{array}{|l} x = \pm 4\\ x \ge -2 \end{array}\Leftrightarrow x=4.[/tex]

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