Answer: ![x^2+y^2=\dfrac{105\pm 18\sqrt6}{2}](https://tex.z-dn.net/?f=x%5E2%2By%5E2%3D%5Cdfrac%7B105%5Cpm%2018%5Csqrt6%7D%7B2%7D)
<u>Step-by-step explanation:</u>
EQ1: x + y = 12 --> x = 12 - y
EQ2: xy = 15
Substitute x = 12-y into EQ2 to solve for y:
(12 - y)y = 15
12y - y² = 15
0 = y² - 12y + 15
↓ ↓ ↓
a=1 b= -12 c=15
![.\ y=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\\\.\quad =\dfrac{-(-12)\pm \sqrt{(-12)^2-4(1)(15)}}{2(1)}\\\\\\.\quad =\dfrac{12\pm \sqrt{144-120}}{2}\\\\\\.\quad =\dfrac{12\pm \sqrt{24}}{2}\\\\\\.\quad =\dfrac{12\pm 2\sqrt{6}}{2}\\\\\\.\quad =6\pm \sqrt{6}](https://tex.z-dn.net/?f=.%5C%20%20y%3D%5Cdfrac%7B-b%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%5C%5C%5C%5C%5C%5C.%5Cquad%20%3D%5Cdfrac%7B-%28-12%29%5Cpm%20%5Csqrt%7B%28-12%29%5E2-4%281%29%2815%29%7D%7D%7B2%281%29%7D%5C%5C%5C%5C%5C%5C.%5Cquad%20%3D%5Cdfrac%7B12%5Cpm%20%5Csqrt%7B144-120%7D%7D%7B2%7D%5C%5C%5C%5C%5C%5C.%5Cquad%20%3D%5Cdfrac%7B12%5Cpm%20%5Csqrt%7B24%7D%7D%7B2%7D%5C%5C%5C%5C%5C%5C.%5Cquad%20%3D%5Cdfrac%7B12%5Cpm%202%5Csqrt%7B6%7D%7D%7B2%7D%5C%5C%5C%5C%5C%5C.%5Cquad%20%3D6%5Cpm%20%5Csqrt%7B6%7D)
Now, let's solve for x:
![xy=15\\\\x(6\pm\sqrt6)=15\\\\x=\dfrac{15}{6\pm\sqrt6}\\\\\\x=\dfrac{15}{6\pm\sqrt6}\bigg(\dfrac{6\pm\sqrt6}{6\pm\sqrt6}\bigg)=\dfrac{6\pm \sqrt6}{2}](https://tex.z-dn.net/?f=xy%3D15%5C%5C%5C%5Cx%286%5Cpm%5Csqrt6%29%3D15%5C%5C%5C%5Cx%3D%5Cdfrac%7B15%7D%7B6%5Cpm%5Csqrt6%7D%5C%5C%5C%5C%5C%5Cx%3D%5Cdfrac%7B15%7D%7B6%5Cpm%5Csqrt6%7D%5Cbigg%28%5Cdfrac%7B6%5Cpm%5Csqrt6%7D%7B6%5Cpm%5Csqrt6%7D%5Cbigg%29%3D%5Cdfrac%7B6%5Cpm%20%5Csqrt6%7D%7B2%7D)
Lastly, find x² + y² :
![y^2=(6\pm \sqrt6)^2\quad \rightarrow \quad y^2=36\pm 12\sqrt6 +6\quad \rightarrow \quad y^2=42\pm 12\sqrt6](https://tex.z-dn.net/?f=y%5E2%3D%286%5Cpm%20%5Csqrt6%29%5E2%5Cquad%20%5Crightarrow%20%5Cquad%20y%5E2%3D36%5Cpm%2012%5Csqrt6%20%2B6%5Cquad%20%5Crightarrow%20%5Cquad%20y%5E2%3D42%5Cpm%2012%5Csqrt6)
![x^2=\bigg(\dfrac{6\pm \sqrt6}{2}\bigg)^2\quad \rightarrow \quad x^2=\dfrac{42\pm 12\sqrt6}{4}\quad \rightarrow \quad x^2=\dfrac{21\pm 6\sqrt6}{2}](https://tex.z-dn.net/?f=x%5E2%3D%5Cbigg%28%5Cdfrac%7B6%5Cpm%20%5Csqrt6%7D%7B2%7D%5Cbigg%29%5E2%5Cquad%20%5Crightarrow%20%5Cquad%20x%5E2%3D%5Cdfrac%7B42%5Cpm%2012%5Csqrt6%7D%7B4%7D%5Cquad%20%5Crightarrow%20%5Cquad%20x%5E2%3D%5Cdfrac%7B21%5Cpm%206%5Csqrt6%7D%7B2%7D)
![x^2+y^2=\dfrac{21\pm 6\sqrt6}{2}+42\pm 12\sqrt6\\\\\\.\qquad \quad = \dfrac{21\pm 6\sqrt6}{2}+\dfrac{84\pm 24\sqrt6}{2}\\\\\\. \qquad \quad = \dfrac{105\pm 18\sqrt6}{2}](https://tex.z-dn.net/?f=x%5E2%2By%5E2%3D%5Cdfrac%7B21%5Cpm%206%5Csqrt6%7D%7B2%7D%2B42%5Cpm%2012%5Csqrt6%5C%5C%5C%5C%5C%5C.%5Cqquad%20%5Cquad%20%3D%20%5Cdfrac%7B21%5Cpm%206%5Csqrt6%7D%7B2%7D%2B%5Cdfrac%7B84%5Cpm%2024%5Csqrt6%7D%7B2%7D%5C%5C%5C%5C%5C%5C.%20%5Cqquad%20%5Cquad%20%3D%20%5Cdfrac%7B105%5Cpm%2018%5Csqrt6%7D%7B2%7D)
$142.20 x .15% = 21.33
$142.20 - 21.33 = $119.87
The ratio of inheritance is
.. sister : brother = 1.25 : 1 = 5 : 4
Then the sister receives 5/(5+4) = 5/9 of the total, $550,000.