Saw this algebra problem in an ad, and don't immediately remember how to solve it:
$\sqrt {x+15} + \sqrt x = 15$
I at least remember that you can't simply square everything, i.e., if a + b = c, it does not necessarily follow that $\ a^2 + b^2 = c^2$.
Of course, it does follow that $(\sqrt {x+15} + \sqrt x)^2 = 15^2 = 225$, but not sure if that's the right direction to go in, or where to go from there if it is. I don't want anyone to solve it for me, but please throw me a hint if you'd be so kind - thanks!
$\endgroup$ 54 Answers
$\begingroup$$$\sqrt { x+15 } +\sqrt { x } =15\\ { \left( \sqrt { x+15 } \right) }^{ 2 }={ \left( 15-\sqrt { x } \right) }^{ 2 }\\$$
$\endgroup$ 3 $\begingroup$$$ x+15=225-30\sqrt { x } +x\\ 30\sqrt { x } =210\\ \sqrt { x } =7\\ x=49$$
Let \begin{align} \sqrt {x+15} + \sqrt x &= 15 \\ \sqrt {x+15} - \sqrt x &= y \end{align}
Multiplying, we get
$(x+15) - x = 15y$
Which simplifies to
$$ y = 1$$
So \begin{align} \sqrt {x+15} + \sqrt x &= 15 \\ \sqrt {x+15} - \sqrt x &= 1 \end{align}
Subtracting, we get
$2 \sqrt x = 14$
Which simplifies to
$\endgroup$ 2 $\begingroup$x = 49
Let $f(x)=\sqrt{x+15}+\sqrt{x}$.
Hence, $f$ is increasing function, which says that our equation has at most one root.
But easy to check that ... is a root. Thus, it's an unique root of our equation.
$\endgroup$ 3 $\begingroup$The key here is that the two radicals are of the form $\sqrt{x+a}$. If you have $$ \sqrt{x+a}+\sqrt{x+b}=k \tag{1} $$ with $a>b$ and $k>0$, then you can multiply both sides by $\sqrt{x+a}-\sqrt{x+b}$, getting $$ (x+a)-(x+b)=k(\sqrt{x+a}-\sqrt{x+b}) $$ so $$ \sqrt{x+a}-\sqrt{x+b}=\frac{a-b}{k}\tag{2} $$ Now look at equations (1) and (2)…
This would also work for equations of the form $$ \sqrt{mx+a}+\sqrt{mx+b}=k $$ or $$ \sqrt{mx+a}-\sqrt{mx+b}=k $$ but not if the coefficients of $x$ are different in the two radicals. In that case, squaring (and hoping for the best) is the way to go.
You could also square: $(\sqrt{x+15}+\sqrt{x})^2=15^2$ so $2x+15+2\sqrt{x(x+15)}=225$ and finally $$ \sqrt{x(x+15)}=105-x $$ Squaring again, $$ x^2+15x=11025-210x+x^2 $$ or $$ 225x=11025 $$ Not really a big deal, but the above methods has smaller figures.
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