1. ## Difficult quadratic equation problem

Hi, I have an 'advanced quadratic' problem which I'm not sure how to solve.

Let pairs $\displaystyle (a, c)$ and $\displaystyle (b, d)$ be roots of the equation $\displaystyle x^2 + ax - b = 0$ and $\displaystyle x^2 + cx + d = 0$ respectively. Find all possible real values for $\displaystyle a, b, c, d.$

Thanks for any help, BG

2. Here's a start.

Since (a,c) are the roots of $\displaystyle x^{2}+ax-b$, then we can write it as

$\displaystyle (x-a)(x-c)=x^{2}+ax-b$

For the other, we can write:

$\displaystyle (x-b)(x-d)=x^{2}+cx+d$

Try equating coefficients and see what you come up with.

Some advice. I assume when you wrote 'complex' you meant the problem was difficult. Do not use the word complex unless you are referring to the non-real numbers. It may add confusion. Just a thought.

3. I think we could use Viete's formula:

$\displaystyle x_1 + x_2 = -\frac{b}{a}$

$\displaystyle x_1 \times x_2 = \frac{c}{a}$

So in this case, for equation 1, with $\displaystyle (a, c)$ = $\displaystyle (x_1, x_2)$ we could do: $\displaystyle x_1 + x_2 = -\frac{b}{a}$ and $\displaystyle x_1 \times x_2 = \frac{c}{a}$ and with $\displaystyle (b, d)$ being $\displaystyle (x_3, x_4)$, $\displaystyle x_3 + x_4 = -\frac{b}{a}$ and $\displaystyle x_3 \times x_4 = \frac{c}{a}$

Just a suggestion, I'm not too sure what to try apart from that.