4
$\begingroup$

Solve: $\sqrt{x-4} + 10 = \sqrt{x+4}$ Little help here? >.<

5 Answers 5

8

There are no real solutions, nor any complex solutions if you use the principal branch of the square root. Squaring both sides and simplifying gives you $20 \sqrt{x-4} = -92$.

EDIT: More generally, for any $a, b \ge 0$, $\sqrt{a + b} \le \sqrt{a} + \sqrt{b}$. Since $(x+4) - (x-4) = 8$, the most $\sqrt{x+4} - \sqrt{x-4}$ can be is $\sqrt{8}$.

  • 1
    The inequality \sqrt{a+b}< \sqrt{a}+\sqrt{b} hasa pretty obvious geometric proof, which leads to the following simple adaptation of your solution: draw the right angle triangle with edges $\sqrt{x-4}, \sqrt{x+4}, \sqrt{8}$, and then by the triangle inequality \sqrt{x+4}< \sqrt{x-4}+\sqrt{8}.2012-09-07
2

We will assume that $x$ ranges over the reals $\ge 4$, to make sure that the square roots are real. Note that $\sqrt{x+4}-\sqrt{x-4}=\frac{(\sqrt{x+4}-\sqrt{x-4})(\sqrt{x+4}+\sqrt{x-4})}{\sqrt{x+4}+\sqrt{x-4}} =\frac{8}{\sqrt{x+4}+\sqrt{x-4}} .$ For $x\ge 4$, $\sqrt{x+4}+\sqrt{x-4}\ge 2\sqrt{2}$. It follows that $\sqrt{x+4}-\sqrt{x-4}\le \dfrac{8}{2\sqrt{2}}=2\sqrt{2}$ for all $x\ge 4$. In particular, $\sqrt{x+4}-\sqrt{x-4}$ cannot be equal to $10$.

0

Questions:

  1. Is the problem written correctly.

  2. Are there restrictions on x?

Something does not seem right in the problem as posed.

Hint: Plot the left hand side and then plot the right hand side and see what it looks like.

  • 0
    That's possibly why the user asked this question.2012-09-07
0

Square both sides, and you get $x - 4 + 20\sqrt{x - 4} + 100 = x + 4$ This simplifies to $20\sqrt{x - 4} = -92$ or just $\sqrt{x - 4} = -\frac{92}{20}$ Since square roots of numbers are always nonnegative, this cannot have a solution.

0

As others have said, there are no solutions within the usual rules. However, once we get to $\sqrt {x-4}=-4.6$ we can remember that square roots can be negative (despite the convention that $\sqrt x \ge 0$). Then we can square and add $4$to find $x=25.16$. Checking, we find $\sqrt {x+4}=5.4, \sqrt{x-4}=-4.6$ and the difference is truly $10$. You can decide if this is better than no answer at all.