Rational Roots of a Quadratic Equation Under Given Conditions

In this article, we will explore the conditions under which the roots of the quadratic equation bc-ax^2 ca-bx ab-c0 are rational. We will focus on the special case where a b c0 and a, b, cinmathbb{Q}.

Step 1: Substituting the Condition

Given that a b c0, we can express c as c-a-b. Substituting this into the equation, we proceed to find the coefficients:

Coefficient of x^2: b(c-a) b(-a-b-a) -2a Coefficient of x: c(a-b) (-a-b)(a-b) -2b Constant term: a(b-c) a(b (-a-b)) ab 2ab 2ab

Thus, the quadratic equation becomes:

-2ax^2-2bx 2ab0

Dividing the entire equation by -2 to simplify:

ax^2 bx-ab0

Step 2: Using the Quadratic Formula

The quadratic formula for roots of the equation ax^2 bx c0 is:

x frac{-b pm sqrt{b^2-4ac}}{2a}

In our case, a a, b b, and c -ab. We need to calculate the discriminant:

D b^2-4ac b^2-4a(-ab) b^2 4a^2b 4a^2

Factoring the discriminant:

D b^2 4a^2b 4a^2 (b^2 4a^2b 4a^2) (b 2a)^2

Since a, binmathbb{Q}, it follows that (b 2a)^2) is a rational number, being the square of a rational number. Thus, the discriminant is a perfect square.

Step 3: Finding the Roots

The roots are given by:

x frac{-b pm (b 2a)}{2a}

This gives us two cases:

First root: x_1 frac{-b (b 2a)}{2a} frac{2a}{2a} 1 Second root: x_2 frac{-b - (b 2a)}{2a} frac{-2b-2a}{2a} -frac{b a}{a}

Since both a and b are rational, x_1 and x_2 are rational numbers.

Therefore, we conclude that the roots of the equation bc-ax^2 ca-bx ab-c0 are indeed rational under the given conditions.

Discriminant Analysis

We can further analyze the discriminant of the simplified equation:

-2ax^2-2bx-2 0 can be rewritten as ax^2 bx c0.

The discriminant for this equation is:

D b^2-4ac b^2-4(-2)(-2) b^2-16ac 4(b^2-4ac)

Since b^2-4ac is rational, 4(b^2-4ac) is the square of a rational number, being four times a perfect square.

Conclusion

Under the conditions that a b c0 and a, b, cinmathbb{Q}, the roots of the equation bc-ax^2 ca-bx ab-c0 are rational. The roots can be explicitly calculated and are rational.

Given the rationality of the coefficients and the perfect square nature of the discriminant, we have successfully demonstrated the rationality of the roots.