How to Find Roots of Quadratic Equation by Splitting Middle Term Using MATLAB

Introduction

Quadratic equations are fundamental in algebra and appear in various fields such as physics, engineering, and economics. Finding their roots is a common task, and while the quadratic formula is widely used, the method of splitting the middle term offers an intuitive alternative. In this blog post, we'll explore how to find roots of quadratic equation by splitting middle term using MATLAB. We'll cover the theory, provide a matlab quadratic equation example, and show a complete matlab quadratic equation implementation.

Understanding Quadratic Equations

A quadratic equation is of the form:

ax² + bx + c = 0, where a ≠ 0.

The roots can be found using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). However, the splitting middle term method involves factoring the quadratic expression into two binomials. This method works well when the roots are rational.

The Splitting Middle Term Method

To split the middle term, we need to find two numbers, p and q, such that:

• p * q = a * c
• p + q = b

Then, we rewrite the middle term bx as px + qx, and factor by grouping. For example, consider 2x² + 7x + 3 = 0. Here, a=2, b=7, c=3. We need p*q=6 and p+q=7. The numbers are 6 and 1. So, 2x² + 6x + x + 3 = 0, which factors to (2x+1)(x+3)=0, giving roots x=-1/2 and x=-3.

Implementing the Method in MATLAB

MATLAB is a powerful tool for mathematical computations. We can write a matlab quadratic equation solver that uses the splitting middle term method. The challenge is to automatically find the pair (p, q) that satisfies the conditions. We can do this by iterating through possible factors of a*c.

Algorithm

1. Input coefficients a, b, c.
2. Compute discriminant D = b² - 4ac. If D < 0, roots are complex; the splitting method may not yield real factors.
3. Find all integer pairs (p, q) such that p*q = a*c and p+q = b.
4. If a pair is found, factor the quadratic and solve for roots.
5. If no integer pair is found, fall back to the quadratic formula.

MATLAB Code Example

Below is a matlab quadratic equation implementation that demonstrates the splitting middle term method:

function roots = splitMiddleTerm(a, b, c)
    % Check if a is zero
    if a == 0
        error('Not a quadratic equation');
    end
    
    % Compute discriminant
    D = b^2 - 4*a*c;
    
    % If discriminant is negative, roots are complex
    if D < 0
        disp('Complex roots. Using quadratic formula.');
        roots = quadraticFormula(a, b, c);
        return;
    end
    
    % Find integer factors of a*c that sum to b
    ac = a * c;
    found = false;
    for p = -abs(ac):abs(ac)
        if p == 0
            continue;
        end
        if mod(ac, p) == 0
            q = ac / p;
            if p + q == b
                found = true;
                break;
            end
        end
    end
    
    if found
        % Factor by grouping: ax^2 + px + qx + c = 0
        % Group as (ax^2 + px) + (qx + c)
        % Factor out common terms
        % For simplicity, we solve using the factored form
        % Roots are -p/a and -q/a? Not exactly; need to factor properly.
        % Actually, if we split bx = px + qx, then ax^2 + px + qx + c = x(ax+p) + (qx+c).
        % But to factor, we need to group as (ax^2 + px) + (qx + c) = x(ax+p) + 1*(qx+c). This doesn't directly factor unless qx+c is multiple of (ax+p).
        % Better approach: use the fact that roots are given by solving (ax+p)(x+? )? Hmm.
        % The splitting method is manual; for code, we can use the quadratic formula after finding p and q to demonstrate.
        % But the requirement is to find roots by splitting middle term. So we need to factor the quadratic.
        % Let's factor: ax^2 + bx + c = (mx + n)(rx + s) where m*r = a, n*s = c, and m*s + n*r = b.
        % Instead of searching p and q, we can search for factors of a and c.
        % But the splitting middle term method specifically finds p and q such that p*q = a*c and p+q = b.
        % Then we can write ax^2 + px + qx + c = 0, and factor by grouping: x(ax+p) + (qx+c). For this to factor, we need qx+c to be a multiple of (ax+p). That is, (qx+c) = k(ax+p). Then k = q/a? Not necessarily integer.
        % Actually, the standard way: after finding p and q, we have ax^2 + px + qx + c = 0. Then we group as (ax^2 + px) + (qx + c) = x(ax+p) + (qx+c). But to factor, we need to express (qx+c) as (ax+p) times something. That something is (q/a) if q/a is such that (q/a)*(ax+p) = qx + (q/a)p. For this to equal qx+c, we need (q/a)p = c, i.e., qp/a = c => qp = ac, which is true. So indeed, (qx+c) = (q/a)(ax+p). Thus, ax^2 + px + qx + c = x(ax+p) + (q/a)(ax+p) = (ax+p)(x + q/a). So roots are x = -p/a and x = -q/a.
        % Wait, check: (ax+p)(x + q/a) = ax^2 + qx + px + pq/a = ax^2 + (p+q)x + pq/a. Since p+q=b and pq=ac, pq/a = c. So yes, it works! So roots are -p/a and -q/a.
        % Therefore, we can directly compute roots as -p/a and -q/a.
        roots = [-p/a, -q/a];
        disp('Roots found by splitting middle term:');
    else
        disp('No integer factors found. Using quadratic formula.');
        roots = quadraticFormula(a, b, c);
    end
end

function roots = quadraticFormula(a, b, c)
    D = b^2 - 4*a*c;
    if D >= 0
        roots = [(-b + sqrt(D))/(2*a), (-b - sqrt(D))/(2*a)];
    else
        roots = [(-b + 1i*sqrt(-D))/(2*a), (-b - 1i*sqrt(-D))/(2*a)];
    end
end

This matlab quadratic equation solution efficiently finds roots when integer factors exist. For non-integer cases, it falls back to the quadratic formula, ensuring robustness.

Example Usage

Let's test with the earlier example: a=2, b=7, c=3.

roots = splitMiddleTerm(2, 7, 3)
% Output: roots = [-0.5000, -3.0000]

This matches our manual calculation. The function correctly identifies p=6, q=1 (or p=1, q=6) and computes roots as -6/2=-3 and -1/2=-0.5.

Advantages and Limitations

The splitting middle term method is excellent for educational purposes and when coefficients are integers. However, it may not find roots if factors are not integers. In such cases, the quadratic formula is more general. Our MATLAB implementation combines both for a complete matlab quadratic equation solver.

Conclusion

In this post, we've shown how to find roots of quadratic equation by splitting middle term using MATLAB. We provided a detailed matlab quadratic equation example and a full matlab quadratic equation implementation. This approach enhances understanding of quadratic equations and demonstrates MATLAB's flexibility. For more complex cases, the quadratic formula remains essential, but the splitting method offers a valuable alternative for simple cases. We hope this matlab quadratic equation calculation guide helps you in your mathematical endeavors.