# FORMATION OF QUADRATIC EQUATION WITH GIVEN ROOTS

If α and β are the two roots of a quadratic equation, then the formula to construct the quadratic equation is

x2 - (α + β)x + αβ  =  0

That is,

x2 - (sum of roots)x + product of roots =  0

If a quadratic equation is given in standard form, we can find the sum and product of the roots using coefficient of x2, x and constant term.

Let us consider the standard form of a quadratic equation,

ax2 + bx + c  =  0

(Here a, b and c are real and rational numbers)

Let α and β be the two zeros of the above quadratic equation.

Then the formula to get sum and product of the roots of a quadratic equation is, Note :

Irrational roots of a quadratic equation occur in conjugate pairs.

That is, if (m + √n) is a root, then  (m - √n) is the other root of the same quadratic equation equation.

## Examples

Example 1 :

Form the quadratic equation whose roots are 2 and 3.

Solution :

Sum of the roots is

=  2 + 3

=  5

Product of the roots is

=  2 x 3

=  6

x2 - (sum of the roots)x + product of the roots  =  0

x2 - 5x + 6  =  0

Example 2 :

Form the quadratic equation whose roots are 1/4 and -1.

Solution :

Sum of the roots is

=  1/4 + (-1)

=  1/4 - 1

=  1/4 - 4/4

=  (1 - 4) / 4

=  -3 / 4

Product of the roots is

=  (1/4) x (-1)

=  -1/4

x2 - (sum of the roots)x + product of the roots  =  0

x2 - (-3/4)x + (-1/4)  =  0

x2 + (3/4)x - 1/4  =  0

Multiply each side by 4.

4x2 + 3x - 1  =  0

Example 3 :

Form the quadratic equation whose roots are 2/3 and 5/2.

Solution :

Sum of the roots is

=  2/3 + 5/2

The least common multiplication of the denominators 3 and 2 is 6.

Make each denominator as 6 using multiplication.

Then,

=  4/6 + 15/6

=  (4 + 15)/6

=  19/6

Product of the roots is

=  2/3 x 5/2

=  5/3

x2 - (sum of the roots)x + product of the roots  =  0

x2 - (19/6)x + 5/3  =  0

Multiply each side by 6.

6x2 - 19x + 10  =  0

Example 4 :

If one root of a quadratic equation (2 + √3), then form the equation given that the roots are irrational.

Solution :

(2 + √3) is an irrational number.

We already know the fact that irrational roots of a quadratic equation will occur in conjugate pairs.

That is,  if (2 + √3) is one root of a quadratic equation, then (2 - √3) will be the other root of the same equation.

So, (2 + √3) and (2 - √3) are the roots of the required quadratic equation.

Sum of the roots is

=  (2 + √3) + (2 - √3)

=  4

Product of the roots is

=  (2 + √3)(2 - √3)

=  22√32

=  4 - 3

=  1

x2 - (sum of the roots)x + product of the roots  =  0

x2 - 4x + 1  =  0

Example 5 :

If α and β be the roots of x2 + 7x + 12  =  0, find the quadratic equation whose roots are

(α + β)2 and (α - β)2

Solution :

Given : α and β be the roots of x2 + 7x + 12  =  0.

Then,

sum of roots  =  -coefficient of x / coefficient of x2

α + β  =  -7 / 1

α + β  =  -7

product of roots  =  constant term / coefficient of x2

αβ  =  12/1

αβ  =  12

Quadratic equation with roots (α + β)2 and (α - β)2 is

x2 - [(α + β)2 + (α - β)2]x + (α + β)2(α - β)=  0 -----(1)

Find the values of (α + β)2 and (α - β)2.

(α + β)=  (-7)2

(α + β)2  =  49

(α - β)2  =  (α + β)2 - 4αβ

(α - β)2  =  (-7)2 - 4(12)

(α - β)2  =  49 - 48

(α - β)2  =  1

So, the required quadratic equation is

(1)-----> x2 - [49 + 1]x + 49 ⋅ 1  =  0

x2 - 50x + 49  =  0

Example 6 :

If α and β be the roots of x2 + px + q  =  0, find the quadratic equation whose roots are

α/β  and  β/α

Solution :

Given : α and β be the roots of x2 + px + q  =  0.

Then,

sum of roots  =  -coefficient of x / coefficient of x2

α + β  =  -p / 1

α + β  =  -p

product of roots  =  constant term / coefficient of x2

αβ  =  q/1

αβ  =  q

Quadratic equation with roots α/β and β/α is

x2 - (α/β + β/α)x + (α/β)(β/α)  =  0

x2 - [α/β + β/α]x + 1  =  0 -----(1)

Find the value of (α/β + β/α).

α/β + β/α  = α2/αβ + β2/αβ

α/β + β/α  = (α+ β2) / αβ

α/β + β/α  = [(α + β)2 - 2αβ] / αβ

α/β + β/α  = (p- 2q) / q

So, the required quadratic equation is

(1)-----> x2 - [(p- 2q) / q]x + 1  =  0

Multiply each side by q.

qx2 - (p- 2q)x + q  =  0 Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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