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Roots And Coefficients Of Quadratic Equation

Relation Between Roots & Coefficients of Quadratic Equation Online Tutoring / Homework Help

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 Alpha and be the roots of the equation, then we have to find the relations of Alpha 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, Alpha = and =

Hence, Alpha + =    +   

Or Alpha + =   =      -   b/a   =   -                  (2)

Again Alpha = x

Or Alpha = =

Or Alpha = = =               (3)

Equations (2) and (3) represent the required relations between roots (that is, Alpha 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 Alpha and , then find Alpha + and Alpha.

Solution:

We know that Alpha + = - = - =

And Alpha = =               (Answer)

Example 2:

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

Solution:

x2 x - 1 = 0               (1)

Let be the other root of the equation (1). Then,

Alpha + = = 1 or = 1 -Alpha

Since Alpha is a root of the equation (1) hence, Alpha 2 Alpha - 1 = 0 or Alpha 2 = Alpha + 1

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

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

Hence, the other root of equation (1) is Alpha3 - 3Alpha.                (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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