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Check the diagonal sum of each candidate.
The Diagonal Sum of a TRUE root-pair must equal to (-b).
The sum c1a2 + c2a1 is called the Diagonal Sum of a root pair.
If the diagonal sum of one of those root pairs gives the right absolute value, but the wrong sign, then the other one is the solution.
Stop computing the Diagonal Sum of other root-sets (or the Sum if a = 1) after it matches (b) or (-b).
Similar(55)
The Generalized Principle of the Golden Section that follows from studying the diagonal sums of the Pascal triangle is a sweeping generalization of these important principles.
In this case, the diagonal sum becomes the sum of the 2 real roots.
Rule for the Diagonal Sum: Given a root pair of a quadratic equation (c1/a1), (c2/a2).
The quadratic equation they satisfy is 15x2 - 4x - 3 = 0. Note that b is the opposite of the diagonal sum as expected.
If the roots are r / s and t / u, then the diagonal sum, ru + ts = - b.
This ru + ts is the "diagonal sum" this method is named after.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com