I'm reading Sawyer's Prelude to Mathematics, here:
I can't understand what's the meaning and application of "condition" here. Also when he gives the example on the cubic equation, stating that the condition is: $$(bc-ad)^2-4(ac-b^2)(bd-c^2)=0$$
I can understand that it is $b^2-4ac=0$ (I hope I'm right with this), I just have no idea on where is the order of the variables inside the parentheses coming from.
1: I noticed that the $b^2-4ac$ can be found here:
$$-b\pm \frac{\sqrt{b^2-4ac}}{2a}$$
Which could be found by solving a general form quadratic equation:
$$ax^2+bx+c=0$$
Then I thought about searching it on the solutions for cubic equations with some help of Mathematica, but I got nothing that was similar to:
$$(bc-ad)^2-4(ac-b^2)(bd-c^2)=0$$
or:
$$a^2d^2-6abcd+4b^3d+4ac^3-3b^2c^2=0$$
With no success. You can see it here: