IMO Solving System Of Equation Problem (Second Attempt)

Solve the following system of equations in real $a,\,b,\,c,\,d$:

$a+b=9$

$ab+c+d=29$

$ad+bc=39$

$cd=18$

Previously we tried the elimination route and we failed.

IMO Optimization Contest Problem: Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xy+xz+yz=4$, and $x, y$ and $z$ are real numbers.

IMO Optimization Contest Problem:

Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xy+xz+yz=4$, and $x, y$ and $z$ are real numbers.

My solution:

From the well-known identity

$(x+y+z)^2=x^2+y^2+z^2+2(xy+xz+yz)$

and the given values for $x^2+y^2+z^2=7$ and $xy+xz+yz=4$, we get:

$(x+y+z)^2=7+2(4)$

China IMO Mock Problem: Given $xy$ is rational number, and $x$ and $y$ are the roots of the equations $6x^2+2015x+8=0$ and $6x^2+2015x+8=0$ respectively. Evaluate $\dfrac{x}{y}$.

Given $xy\ne 1$, and $x$ and $y$ are the roots of the equations $6x^2+2015x+8=0$ and $6x^2+2015x+8=0$ respectively. Evaluate $\dfrac{x}{y}$.

Solution:

Since the coefficients of $6x^2+2015x+8=0$ are the same as the coefficients of $8y^2+2015y+6$ in reversed order, and we're told that $x$ is the root of the equation $6x^2+2015x+8=0$ while $y$ is the root of the equation  $8y^2+2015y+6$.