    # Formation Of Quadratic Equation With Given Roots Let a quadratic equation be ax2 + bx + c = 0 [a ≠ 0]. Let the roots of this equation be and �.

Hence + � = and � = .

Now, ax2 + bx + c = 0 or x2 + x + = 0 [Since a ≠ 0]

Or x2 � ( + �)x + � =0[Since + � = and � = ]

Or x2 � (sum of the roots)x + product of the roots = 0            (1)

Equation (1) is known as the formula (1) which is used for the formation of a quadratic equation when its roots are given.

For example, if we have to form a quadratic equation whose roots are 2 and -3. Then by formula (1) we get the required equation as given below:

x2 � [2 + (-3)]x + 2.(-3) = 0 or x2 � (-1)x + (-6) = 0 or x2 + x � 6 = 0.

Example 1:

If and � be the roots of the equation x2 + 3x + 4 = 0, then find the equation whose roots are ( + �)2 and ( � �)2.

Solution:

x2 + 3x + 4 = 0            (1)

Given that the roots of equation (1) are and �. Hence + � = -3 and � = 4.

Now, the sum of the roots of the required equation = ( + �)2 + ( � �)2

= (-3)2 + [( + �)2 - 4 �]

= 9 + [(-3)2 � 4.4]

= 9 + 9 � 16 = 2

And the product of the roots of the required equation = ( + �) 2.( � �)2

= (-3)2. [(-3)2 � 4.4]

= 9.(-7) = -63

Hence, the required equation is,

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

or x2 � 2x + (-63) = 0

or x2 � 2x - 63 = 0            (Answer)

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