. . Vieta’s formula for Quadratic Equations Let α and β be the roots of the quadratic equation ax bx . Then ax bx + = a x ax ) = ) + ( ) = β β αβ .
Equating the coefficients of like powers, we see that β = − b a and αβ = c a . So a quadratic equation whose roots are α and β is x β αβ ; that is, a quadratic equation with given roots is, x − (sum of the roots) x + product of the roots = . ... ( ) Note The indefinite article a is used in the above statement.
In fact, if P x ( ) = is a quadratic equation whose roots are α and β , then cP x ( ) is also a quadratic equation with roots α and β for any non-zero constant c . - - Theory of Equations In earlier classes, using the above relations between roots and coefficients we constructed a quadratic equation, having α and β as roots. In fact, such an equation is given by ( ). For instance, a quadratic equation whose roots are and is given by x = .
Further we construct new polynomial equations whose roots are functions of the roots of a given polynomial equation; in this process we form a new polynomial equation without finding the roots of the given polynomial equation. For instance, we construct a polynomial equation by increasing the roots of a given polynomial equation by two as given below. Example . If α and β are the roots of the quadratic equation17 x , construct a quadratic equation whose roots are α + 2and β + .
Since α and β are the roots of x , we have α β = − and αβ = − . We wish to construct a quadratic equation with roots α + and β + .Thus, to construct such a