Introduction to Quadratic Equations with Given Roots

Quadratic equations are polynomial equations of the second degree, which can be written in the standard form: ax2 bx c 0. The roots of such equations are the values of x that satisfy the equation. In this article, we will explore how to find the general form of a quadratic equation given its roots.

Determining the Quadratic Equation from Given Roots

Let's consider the example where the roots of a quadratic equation are 2/3 and -1/2. To find the quadratic equation, we can use the fact that for a quadratic equation ax2 bx c 0, the sum of the roots is given by -b/a and the product of the roots is given by c/a.

Step 1: Finding the Sum and Product of the Roots

The sum of the roots is calculated as follows:

S 2/3 (-1/2) 1/6

The product of the roots is calculated as follows:

P (2/3)(-1/2) -1/3

Step 2: Forming the Quadratic Equation

Given the sum and product of the roots, we can form the quadratic equation in the standard form:

[ax^2 - Sx P 0]

Substituting the values of S and P, we get:

[ax^2 - (1/6)x - 1/3 0]

To eliminate the fractions, we can multiply through by 6:

[6x^2 - x - 2 0]

Verification of the Solution

To verify the solution, we can solve for x using the quadratic formula:

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

Substituting a 6, b -1, and c -2, we get:

[x frac{-(-1) pm sqrt{(-1)^2 - 4(6)(-2)}}{2(6)}]

[x frac{1 pm sqrt{1 48}}{12}]

[x frac{1 pm sqrt{49}}{12}]

[x frac{1 pm 7}{12}]

Solving for the two roots:

[x frac{8}{12} frac{2}{3}]

[x frac{-6}{12} -frac{1}{2}]

This verifies that the roots are indeed 2/3 and -1/2.

General Form of Quadratic Equations with Given Roots

Given that a quadratic equation can have infinitely many variations while maintaining the same roots, the general form of a quadratic equation with roots 2/3 and -1/2 can be written as:

[k(x - 2/3)(x 1/2) 0]

Expanding this, we get:

[k(x^2 - 1/6x - 2/3) 0]

Multiplying through by 6 to clear the fractions:

[6kx^2 - kx - 2k 0]

Here, k can be any non-zero real number.

Conclusion

Quadratic equations with specific roots can be determined using the sum and product of the roots. By understanding the relationship between the roots and the coefficients of the equation, we can construct a quadratic equation that satisfies the given conditions. This knowledge is fundamental in various fields, including algebra, physics, and engineering, where quadratic equations are frequently encountered.