Relation between Roots and Coefficients of a Quadratic Equation

A quadratic equation is a polynomial equation of second degree, i.e. it contains the variable raised to the power of two. A quadratic equation can be expressed in general form as:

$ax^2 + bx + c = 0$

where a, b, and c are coefficients of the equation and x is the variable. The roots or solutions of a quadratic equation can be found by using the quadratic formula:

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

In this article, we will explore the relationship between the roots and coefficients of a quadratic equation.

The Product of Roots

One of the most important relations between the roots and coefficients of a quadratic equation is the product of roots. The product of the roots of a quadratic equation is equal to the constant term divided by the coefficient of the squared term.

$r_1 \cdot r_2 = \frac{c}{a}$

where $r_1$ and $r_2$ are the roots.

For example, consider the quadratic equation:

$2x^2 - 6x + 4 = 0$

The roots of this equation can be found using the quadratic formula as:

$x = \frac{6 \pm \sqrt{6^2-4(2)(4)}}{2(2)}$

which simplifies to:

$x_1 = 1, \quad x_2 = 2$

We can verify that the product of the roots is the ratio of the constant term to the coefficient of the squared term:

$r_1 \cdot r_2 = (1) \cdot (2) = 2$
$\frac{c}{a} = \frac{4}{2} = 2$

Indeed, we can see that the relation holds.

The Sum of Roots

Another important relation between the roots and coefficients of a quadratic equation is the sum of the roots. The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term, divided by the coefficient of the squared term.

$r_1 + r_2 = -\frac{b}{a}$

For the same example quadratic equation:

$2x^2 - 6x + 4 = 0$

The sum of the roots can be found using the formula as:

$r_1 + r_2 = \frac{-(-6)}{2} = 3$

Conclusion

In conclusion, the roots and coefficients of a quadratic equation are closely related. The product of the roots is equal to the ratio of the constant term to the coefficient of the squared term, and the sum of the roots is equal to the negation of the coefficient of the linear term, divided by the coefficient of the squared term. These relations are important in solving quadratic equations and in understanding the properties of quadratic functions.

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二次方程式における解と係数の関係[JA]