PlanetMath: properties of quadratic equation

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properties of quadratic equation

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The quadratic equation

$\displaystyle ax^2\!+\!bx\!+\!c = 0$

$\displaystyle x^2\!+\!px+\!q\! = 0$

with rational, real, algebraic or complex coefficients ( $a \neq 0$ ) has the following properties:

It has in $\mathbb{C}$ two roots (which may be equal), since the complex numbers form an algebraically closed field containing the coefficients.
The sum of the roots is equal to $-\frac{b}{a}$ , i.e.
The product of the roots is equal to $\frac{c}{a}$ , i.e.

Corollary. If the leading coefficient and the constant term are equal, then the roots are inverse numbers of each other.

Without solving the equation, the value of any symmetric polynomial of the roots can be calculated.

Example. If one has to calculate $x_1^3\!+\!x_2^3$ , when and are the roots of the equation $x^2\!-\!4x\!+\!9 = 0$ , so $x_1\!+\!x_2 = 4$ and . Because

$\displaystyle (x_1\!+\!x_2)^3 = x_1^3\!+\!3x_1^2x_2\!+\!3x_1x_2^2\!+\!x_2^3 = (x_1^3\!+\!x_2^3)\!+\!3x_1x_2(x_1\!+\!x_2),$

we obtain

$\displaystyle x_1^3\!+\!x_2^3 = (x_1\!+\!x_2)^3\!-\!3x_1x_2(x_1\!+\!x_2) = 4^3\!-\!3\cdot 9\cdot 4 = -44.$

Note. If one wants to write easily a quadratic equation with rational roots, one could take such one that the sum of the coefficients is zero (then one root is always 1). For instance, the roots of the equation $5x^2\!+\!11x\!-\!16 = 0$ are 1 and $-\frac{16}{5}$ .

"properties of quadratic equation" is owned by pahio.

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quadratic equation

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Cross-references: symmetric polynomial, equation, inverse numbers, leading coefficient, product, sum, field, algebraically closed, complex numbers, roots, properties, coefficients, complex, real, rational, quadratic equation
There are 6 references to this entry.

This is version 9 of properties of quadratic equation, born on 2004-08-30, modified 2006-01-12.
Object id is 6115, canonical name is PropertiesOfQuadraticEquation.
Accessed 4769 times total.

Classification:

AMS MSC:

12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )

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