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

**x ^{2}^{ }- (α + β)x + αβ = 0**

That is,

**x ^{2}^{ }- (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 x^{2}, x and constant term.

Let us consider the standard form of a quadratic equation,

ax^{2} + 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.

**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

Formation of quadratic equation :

x^{2} - (sum of the roots)x + product of the roots = 0

x^{2} - 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

Formation of quadratic equation :

x^{2} - (sum of the roots)x + product of the roots = 0

x^{2} - (-3/4)x + (-1/4) = 0

x^{2} + (3/4)x - 1/4 = 0

Multiply each side by 4.

4x^{2} + 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

Formation of quadratic equation :

x^{2} - (sum of the roots)x + product of the roots = 0

x^{2} - (19/6)x + 5/3 = 0

Multiply each side by 6.

6x^{2} - 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)

= 2^{2} - √3^{2}

= 4 - 3

= 1

Formation of quadratic equation :

x^{2} - (sum of the roots)x + product of the roots = 0

x^{2} - 4x + 1 = 0

**Example 5 : **

If α and β be the roots of x^{2} + 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 x ^{2} + 7x + 12 = 0.**

**Then, **

**sum of roots = -coefficient of x / coefficient of x ^{2}**

**α + β ****= -7 / 1**

**α + β ****= -7**

**product of roots = constant term / coefficient of x ^{2}**

**αβ ****= 12/1**

**αβ ****= 12**

**Quadratic equation with roots ****(α + β) ^{2} and (α - β)^{2 }is**

**x ^{2} - [**(α + β)

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

(α + β)^{2 }= (-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)-----> x ^{2} - [49 + 1]x + 49 **⋅ 1

**x ^{2} - 50x + 49**

**Example 6 : **

If α and β be the roots of x^{2} + px + q = 0, find the quadratic equation whose roots are

α/β and β/α

**Solution :**

**Given : α and β be the roots of x ^{2} + px + q = 0.**

**Then,**

**sum of roots = -coefficient of x / coefficient of x ^{2}**

**α + β ****= -p / 1**

**α + β ****= -p**

**product of roots = constant term / coefficient of x ^{2}**

**αβ ****= q/1**

**αβ ****= q**

**Quadratic equation with roots ****α/β and β/****α**^{ }is

**x ^{2} - (**α/β + β/α)x + (α/β)(β/α) = 0

**x ^{2} - [**α/β + β/α]x + 1 = 0 -----(1)

Find the value of (α/β + β/α).

α/β + β/α = α^{2}/αβ + β^{2}/αβ

α/β + β/α = (α^{2 }+ β^{2}) / αβ

α/β + β/α = [(α^{ }+ β)^{2 }- 2αβ] / αβ

α/β + β/α = (p^{2 }- 2q) / q

So, the required quadratic equation is

**(1)-----> x ^{2} - [**(p

**Multiply each side by q. **

**qx ^{2} - **(p

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