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The given equation is

$$ \sqrt{x} + \sqrt{x+16} = 3$$

What is the range of the solution?

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3  
Note $x$ cannot be negative, so $\sqrt{x+16}\ge4$. –  David Mitra Aug 10 '12 at 1:30
    
How do you usually solve equations that have square roots in them? Try that and see what happens. –  Francis Adams Aug 10 '12 at 1:33

3 Answers 3

up vote 3 down vote accepted

Clearly existence of $\sqrt x\implies x\geq 0$ for $x\in \Bbb R$.Thus, $\sqrt {x+16}$ is atleast $4\implies \sqrt x+\sqrt {x+16}\geq 4$ for $x\in \Bbb R\implies$ no real solution for $\sqrt x+\sqrt {x+16}=3$

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Try this. Be aware $x\ge 0$ or we are dead on the spot. We begin with this.

$$\sqrt{x + 16} = 3 -\sqrt{x}$$ Squaring gives $$x + 16 = 9 - 6\sqrt{x} + x. $$ Now cancel to get $$6\sqrt{x} = -7. $$ This does not look so good. I think it's devoid of real solutions.

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There aren't any solutions. –  ncmathsadist Aug 10 '12 at 1:36

Take square of both sides and get $$ 2x+16+2\sqrt{x(x+16)}=9 $$ We thus get $$ \sqrt{x(x+16)}=-x-\frac{7}{2}. $$ Take square of both sides and get $$ x(x+16)=x^2+7x+\frac{49}{4}. $$ This is a quadratic equation in $x$, so the rest is easy.

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3  
But then you have to plug the solutions you find into the original equation to see if they are real or were introduced by the squaring. –  Ross Millikan Aug 10 '12 at 1:42
    
You are right. I noticed David Mitra's comment above. –  Michel Aug 10 '12 at 4:52

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