Constructing a Quadratic Equation from Given Roots

In mathematics, especially in algebra, finding the quadratic equation given its roots is a classic problem. This article will explore the step-by-step process of constructing a quadratic equation from the roots 2/3 and 4. We will use both Vieta's formulas and the general method to derive the same quadratic equation.

Using Vieta's Formulas

Vi?ta's formulas provide a straightforward method to derive a quadratic equation given its roots. If ax2 bx c 0 has roots q and r, then:

-b/a q r c/a qr

Given the roots q 2/3 and r 4, we can calculate:

-b/a 2/3 4 14/3 c/a (2/3) * 4 8/3

Choosing a 3, we get:

b -14 c 8

Thus, the quadratic equation is:

3x2 - 14x 8 0

This can be further simplified using polynomial factorization techniques to get:

3x - 2x - 4 0

Alternative Method Using Polynomial Factors

An alternative method involves expressing the quadratic equation as a product of its roots. Given the roots 2/3 and 4, the equation can be written as:

(x - 2/3)(x - 4) 0

Expanding this product, we get:

x2 - (2/3 4)x (2/3 * 4) 0

which simplifies to:

x2 - 14/3x 8/3 0

Multiplying through by 3 to clear the fractions, we obtain:

3x2 - 14x 8 0

This is the same quadratic equation as before, confirming the correctness of the method.

Generalization

To generalize the problem, consider the roots 2/3 and 4 and let a ≠ 0. The general form of the quadratic equation is:

ax2 - a(2/3 4)x a(2/3 * 4) 0

which simplifies to:

ax2 - 14ax/3 8a/3 0

or

3ax2 - 14ax 8a 0

This is a family of quadratic equations parameterized by a.

Key Takeaways

The problem of finding the quadratic equation given its roots can be solved using Vieta's formulas or polynomial factorization. Both methods yield the same result. Understanding these methods is crucial for solving a wide range of algebraic problems.

Keywords: quadratic equation, roots, Vieta's formulas