Finding the Quadratic Equation with Given Roots

Understanding the relationship between a quadratic equation and its roots is a fundamental aspect of algebra. In this article, we will demonstrate how to find the quadratic equation with roots (r_1 frac{1}{2}) and (r_2 frac{2}{3}).

Step-by-Step Guide

To find a quadratic equation given its roots, we can use the general form of a quadratic equation:

[ax^2 bx c 0]

The sum of the roots (r_1) and (r_2) can be expressed as:

[r_1 r_2 frac{1}{2} frac{2}{3}]

The product of the roots (r_1) and (r_2) can be expressed as:

[r_1 cdot r_2 frac{1}{2} cdot frac{2}{3}]

Calculating the Sum and Product of the Roots

Let's calculate the sum of the roots:

[frac{1}{2} frac{2}{3} frac{3}{6} frac{4}{6} frac{7}{6}]

Next, let's calculate the product of the roots:

[frac{1}{2} cdot frac{2}{3} frac{2}{6} frac{1}{3}]

Formulating the Quadratic Equation

Using the sum and product of the roots, we can write the quadratic equation in the form:

[x^2 - left(r_1 r_2right)x r_1 cdot r_2 0]

Substituting the values we found:

[x^2 - frac{7}{6}x frac{1}{3} 0]

Eliminating Fractions

To clear the fractions, we can multiply the entire equation by 6:

[6left(x^2 - frac{7}{6}x frac{1}{3}right) 0]

This simplifies to:

[6x^2 - 7x 2 0]

Understanding the General Form of Quadratic Equations

The general form of a quadratic equation can be written as:

[kx^2 - 7x 2 0]

Here, (k) is any real number except zero. This means that there are infinitely many quadratic equations with the same roots. One common form is with (k 6), giving us:

[6x^2 - 7x 2 0]

Conclusion

The quadratic equation with roots (r_1 frac{1}{2}) and (r_2 frac{2}{3}) is:

[6x^2 - 7x 2 0]

This article demonstrates the steps to find a quadratic equation given its roots and illustrates the importance of the sum and product of the roots in determining the equation.