Vieta’s formula for Quadratic Equations

Vieta’s Formulae and Formation of Polynomial Equations

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 formula for Quadratic Equations

Let α and β be the roots of the quadratic equation ax² + bx + c = 0. Then ax² + bx + c = a ( x − α )( x − β ) = ax² − a (α + β ) x + a (αβ ) = 0. 

Equating the coefficients of like powers, we see that

α + β = −b/a and αβ = c/a.

So a quadratic equation whose roots are α and β is x² − (α + β )x + αβ = 0 ; that is, a quadratic equation with given roots is,

x² âˆ’ (sum of the roots) x + product of the roots = 0. (1)

Note

The indefinite article a is used in the above statement. In fact, if P(x) = 0 is a quadratic equation whose roots are α and β , then cP(x) is also a quadratic equation with roots α  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 α and β 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 x² − 7x +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.

Example 3.1

If α and β are the roots of the quadratic equation17x² + 43x − 73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2 .

Solution

Since α and β are the roots of 17x² + 43x - 73 = 0 , we have α + β =  -43/17 and αβ = -73/17 

We wish to construct a quadratic equation with roots areα + 2 and β + 2 .Thus, to construct such a quadratic equation, calculate,

the sum of the roots  =  α + β + 4  =  [-43/17] + 4 =  [25/17] and

the product of the roots = αβ + 2(α + β ) + 4 =  (-73/17) + 2 ( -43 /17) + 4 = -91/17 .

Hence a quadratic equation with required roots is x² – (25/17) x – (91/17) = 0

Multiplying this equation by 17, gives 17x² - 25x - 91 = 0

which is also a quadratic equation having roots  α + 2 and  β + 2 .

Example 3.2

If α and β are the roots of the quadratic equation 2x² − 7x +13 = 0 , construct a quadratic equation whose roots are α² and β².

Solution

Since α and β are the roots of the quadratic equation, we have α + β = 7/2 and αβ = 13/2

Thus, to construct a new quadratic equation,

Sum of the roots = α² + β² = (α + β)² - 2αβ = -3/4.

Product of the roots  =  α² β²  = (αβ)²  = 169/4

Thus a required quadratic equation is x² + (¾)x + (169/4) = 0 . From this we see that

4x²  + 3x +169  = 0

is a quadratic equation with roots α² and β².

Remark

In Examples 3.1 and 3.2, we have computed the sum and the product of the roots using the known α + β and αβ . 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 α and β even after completing the task.