Getting conditions for a dense algebraic second degree equation

I have a dense algebraic second degree equation, with five parameters in each coefficiente.

I'd like to get conditions about positivity of the solutions.

Do you know if this is factible in MATLAB?

Thank you in advance.

[Torsten]

So you have a quadratic polynomial

f(x) = a*x^2+b*x+c

where a, b and c depend on 5 parameters a = a(p1a,p2a,p3a,p4a,p5a), b= b(p1b,p2b,p3b,p4b,p5b) and c=c(p1c,p2c,p3c,p4c,p5) and you want to know when

x(1) = -b/(2a) + 1/(2*a)*sqrt(b^2- 4ac) and

x(2) = -b/(2a) - 1/(2*a)*sqrt(b^2- 4ac)

are both positive ?

[Noemi ZM]

Yes, that's it!! In fact, the 5 parameters are the same for each coefficient, and I want to get a law for exactly one of this in relation to the others s.t. the solution are positive.

[Torsten]

If b^2-4*a*c is positive, a sufficient condition is

b/(2*a) < 1/(2*a)*sqrt(b^2- 4ac) < -b/2a

[Noemi ZM]

Thank you, Torsten, but the coefficients are really hard, so I'd like a program that can isolate the parameter p1 for me, could you understand?

Answer 1

[Ravi]

Hi Noemi ZM,

Let us assume that the coefficients of the algebraic equation are governed by the parameters “p1, p2, p3, p4, and p5”. The algebraic equation be , where the coefficients are all some functions of the parameters, say,

a = computeA(p1, p2, p3, p4, p5);
b = computeB(p1, p2, p3, p4, p5);
c = computeC(p1, p2, p3, p4, p5);

Please note that, when the two solutions of a second-degree equation are to be positive, the following conditions should hold,

1. The discriminant value should be greater than or equal to zero. That implies, .
2. The product of the two solutions should be greater than zero. That implies, c and a should have same sign.
3. The sum of the two solutions should be greater than zero. That implies, b and a should have the opposite signs.

We can design a function in MATLAB, that does the following.

1. Input the five parameters.
2. Compute the coefficients.
3. Verify if the conditions to hold are satisfied.
4. Return true if all conditions are met, otherwise false.

function result = areSolutionsPositive(p1, p2, p3, p4, p5)
    conditionsSatisfied = 0;

    % Compute the coefficients
    a = computeA(p1, p2, p3, p4, p5);
    b = computeB(p1, p2, p3, p4, p5);
    c = computeC(p1, p2, p3, p4, p5);

    % Compute discriminant(delta)
    delta = b * b - 4 * a * c;

    % Condition 1: delta is greater than or equal to zero
    if delta >= 0
        conditionsSatisfied = conditionsSatisfied + 1;
    end

    % Condition 2: c and a should have the same sign
    % In other words, c * a should be greater than zero
    if c * a > 0
        conditionsSatisfied = conditionsSatisfied + 1;
    end

    % Condition 3: b and a should have opposite signs
    if b * a < 0
        conditionsSatisfied = conditionsSatisfied + 1;
    end

    % Verify if all the three conditions are satisfied
    if conditionsSatisfied == 3
        result = true;
    else
        result = false;
    end
end

Hope this answer helps you the solve the issue you are facing.

[Noemi ZM]

Thank you, Ravi, varying the parameters in a mesh I hope to get important clues with your idea.