If $a^2+a b+b^2=40$ and $a^2-\sqrt{a b}+b=5$, then find $a^2+\sqrt{a b}+b$

I was given this problem to solve with elementary methods (High School level).

Knowing that $$\begin{align} a^2+a b+b^2 &=40 \\ a^2-\sqrt{a b}+b &=\phantom{0}5 \end{align}$$ find $$a^2+\sqrt{a b}+b$$

I tried to look for $\sqrt{a b}$, since the requested quantity is

$$a^2+\sqrt{a b}+b=(a^2-\sqrt{a b}+b)+2\sqrt{ab}=5 + 2\sqrt{ab}$$ So I set $$x=a^2;\;y=\sqrt{ab};\;b=z$$ and the system became $$x+y^2+z^2=40;\;x-y-z=5$$ subtracting the two equations I got $$z^2-z+y^2+y-35=0$$ which has one real solution when the discriminant is zero.

That is

$1-4(y^2+y-35)=0$

and then

$y=\frac{1}{2} \left(-1\pm\sqrt{142}\right)$

and finally $$a^2+\sqrt{a b}+b=4\pm\sqrt{142}$$

I know that there are other solutions because I've found them with Wolfram Mathematica, but I couldn't find them with elementary methods.

Any help will be appreciated.

Solution 1:

We could try to "brute force" it at the high school level. Start with the second equation $$a^2 - \sqrt{ab} + b = 5$$ $$a^2 + b -5 = \sqrt{ab}$$ $$a^4 + b^2 + 25 + 2a^2 b -10a^2 -10b = ab$$ $$b^2 + ( 2a^2 -10 -a) b + (a^4 -10a^2 +25) =0$$ $$b = \frac{-(2a^2 -10 -a) \pm \sqrt{( 2a^2 -10 -a)^2 - 4(a^4 -10a^2 +25))}}{2}$$ $$b = \frac{-(2a^2 -10 -a) \pm \sqrt{a (-4 a^2 + a + 20)}}{2}$$ Now let's just call $b=x \pm \sqrt{y}.$ $$b^2 = x^2 \pm 2x\sqrt{y} + y$$ $$ab = ax \pm a\sqrt{y}$$ Plug this into the second equation: $$a^2 +ab + b^2 =40$$ $$a^2 + ax \pm a\sqrt{y} + x^2 \pm 2x\sqrt{y} + y =40 $$ $$(\pm a \pm 2x)\sqrt{y}=40 -a^2 - ax - x^2 -y$$ $$(a^2 \pm 2xa + 4x^2)y=(40 -a^2 - ax - x^2 -y)^2$$ Now plug in for $x, y$ and solve for $a$. What you end up with is: $$-225 + 950 a - 290 a^2 - (255 a^3)/2 - (53 a^4)/4 - (45 a^5)/2 + 14 a^6 + 2 a^7 - a^8 =0$$ This is not tractable, and it is likely the problem has a typo.

If you type in Mathematica

Solve[{a^2 + a*b + b^2 == 40, \[Sqrt](a*b) + 5 == a^2 + b}, {a, b}]

It will spit out 3 gross looking expressions possible answers. If you then evaluate these for $a^2 + b + \sqrt{ab}$

a^2 + b + Sqrt[a b] /. %OUTPUT NUMBER

Then the resulting expression doesn't simplify nicely. (Which is not a guarantee; Mathematica's simplify function is far from perfect).