$a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0 $ Let $a,\,b,\,c,\,d$ be distinct real numbers and $a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0$. Then find the value of $a+b+c+d$.
I  could only get $2$ equations that 
$a=2c-b$ and $c=2a-d$.

Let the two polynomials be $$
p(x) = x^2 - 2cx -5d \\
q(x) = x^2 - 2ax - 5b
$$
You also know that $$
p(x) = (x-a)(x-b) = x^2 - (a+b)x + ab \\
q(x) = (x-c)(x-d) = x^2 - (c+d)x + cd
$$
You might try playing around with these two forms.  For example, you can take the product of the polynomials and equate the coefficients for each power of $x$.
