    # Roots And Coefficients Of Quadratic Equation Let us take the general form of a quadratic equation:

ax2 + bx + c = 0            (1)

where a(≠ 0) is the coefficient of x2 , b is the coefficient of x and c is a constant term. If and � be the roots of the equation, then we have to find the relations of and � with a, b and c.

Since a ≠ 0, hence multiplying both sides of (1) by 4a we get,

4a2x2 + 4abx + 4ac = 0 or (2ax)2 + 2.2ax.b + b2 � b2 + 4ac = 0

Or, (2ax + b)2 = b2 � 4ac 2ax + b = b2 � 4ac x = Hence, the roots of (1) are Let, = and � = Hence, + � = + Or + � = =      -   b/a   =   - (2)

Again � = x Or � = = Or � = = = (3)

Equations (2) and (3) represent the required relations between roots (that is, and �) and coefficients (that is, a, b and c) of equation (1).

Example 1:

If the roots of the equation 2x2 - 9x - 3 = 0 be and �, then find + � and �.

Solution:

We know that + � = - = - = And � = = (Answer)

Example 2:

If one root of the quadratic equation x2 � x - 1 = 0 is a, prove that its other root is 3 - 3 .

Solution:

x2 � x - 1 = 0               (1)

Let � be the other root of the equation (1). Then, + � = = 1 or � = 1 - Since is a root of the equation (1) hence, 2 - 1 = 0 or 2 = + 1

Now, 3 - 3 = . 2 - 3 = ( + 1) - 3 [Since 2 = + 1]

= 2 + - 3 = + 1 - 2 = 1 � = �       [Since � = 1 � ]

Hence, the other root of equation (1) is 3 - 3 .                (Proved)

Example 3:

If a2 = 5a � 3 and b2 = 5b � 3, (a ≠ b), find the quadratic equation whose roots are and .

Solution:

Given (a ≠ b) and a2 = 5a � 3 and b2 = 5b � 3, hence it is clear that a and b are the roots of the equation x2 = 5x � 3 or x2 � 5x + 3 = 0.

Hence, a + b = - = 5 and ab = = 3.

Now, the sum of the roots of the required equation

= + = = = = = And the product of the roots of the required equation = . = 1.

Hence, the required equation is x2 x + 1 = 0 or 3x2 � 19x + 3 = 0.                (Answer)

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