Vieta's formulae relate the coefficients of a polynomial to sums and products of its roots. Vieta was a French mathematician whose work on polynomials paved the way for modern algebra.

Vieta's formulae relate the coefficients of a polynomial to sums
and products of its roots. Vieta was a French mathematician whose work on
polynomials paved the way for modern algebra.

**Vietaâ€™s formula for Quadratic Equations**

Let *Î±** *and *Î²** *be the roots of the quadratic equation *ax*^{2} + *bx *+ *c *= 0. Then *ax*^{2} + *bx *+ *c *= *a *( *x *âˆ’ *Î±** *)( *x *âˆ’ *Î²** *) = *ax*^{2} âˆ’ *a *(*Î±** *+ *Î²** *) *x *+ *a *(*Î±Î²** *) = 0.

Equating the coefficients of like powers, we see that

*Î± + Î² = âˆ’b/a *and* Î±Î² = c/a.*

So **a **quadratic equation whose roots are *Î±** *and *Î²** *is *x*^{2} âˆ’ (*Î±** *+ *Î²** *)*x *+ *Î±Î²** *= 0 ; that is, **a **quadratic
equation with given roots is,

*x*^{2} âˆ’ (sum of the roots) *x *+ product of the roots
= 0. (1)

The indefinite article **a **is used in the above statement. In
fact, if *P*(*x*) =
0 is a quadratic equation whose roots are *Î±** *and *Î²** *, then *cP*(*x*) is also a quadratic equation with
roots *Î±** *and *Î²** *for any non-zero
constant *c*.

In earlier classes, using the above relations between roots and
coefficients we constructed a quadratic equation, having *Î±** *and *Î²** *as roots. In fact,
such an equation is given by (1). For instance, a quadratic equation whose
roots are 3 and 4 is given by *x*^{2} âˆ’ 7*x *+12 = 0.

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.

If *Î± *and *Î² *are the roots of the quadratic
equation17*x*^{2} + 43*x *âˆ’ 73 = 0 , construct a quadratic equation whose
roots are *Î± *+ 2 and *Î² *+ 2 .

Since *Î± *and *Î² *are the roots of 17*x** ^{2}* + 43

We wish to construct a quadratic equation with roots are*Î± *+
2 and *Î² *+ 2 .Thus, to construct such a quadratic equation, calculate,

the sum of the roots =
Î± + Î² + 4 = [-43/17] + 4 = [25/17] and

the product of the roots = Î±Î² + 2(Î± + Î² ) + 4 = (-73/17) +
2 ( -43 /17) + 4 = -91/17 .

Hence a quadratic equation with required roots is *x*^{2} â€“ (25/17) *x* â€“
(91/17) = 0

Multiplying this equation by 17, gives 17*x** ^{2}* - 25

which is also a quadratic equation having roots *Î± *+
2 and *Î² *+ 2 .

If *Î± *and *Î² *are the roots of the quadratic equation 2*x*^{2} âˆ’ 7*x *+13 = 0 ,
construct a quadratic equation whose roots are *Î±*^{2} and *Î²*^{2}.

Since *Î± *and *Î² *are the roots of the quadratic
equation, we have *Î± *+ *Î² *= 7/2 and Î±Î² = 13/2

Thus, to construct a new quadratic equation,

Sum of the roots = Î±^{2} + Î²^{2} = (Î± + Î²)^{2} - 2Î±Î² = -3/4.

Product of the roots =
Î±^{2} Î²^{2} = (Î±Î²)^{2} = 169/4

Thus a required quadratic equation is x^{2} + (Â¾)*x* +
(169/4) = 0 . From this we see that

4*x** ^{2}* + 3

is a quadratic equation with roots *Î±** ^{2}* and

In Examples 3.1 and 3.2, we have computed the sum and the
product of the roots using the known *Î±** *+ *Î²** *and *Î±Î²** *. In this way we can construct quadratic equation with desired
roots, provided the sum and the product of the roots of a new quadratic
equation can be written using the sum and the product of the roots of the given
quadratic equation. We note that we have not solved the given equation; we do
not know the values of *Î±** *and *Î²** *even after completing the task.

Tags : Theory of Equations , 12th Mathematics : UNIT 3 : Theory of Equations

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