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Condition for one common root

Let α be the common root of the given equations a1 x2 + b1 x + c1 = 0 and a2 x2 + b2 x + c2 = 0.
 
Then, a1 α2 + b1 α + c1 = 0 and a2 α2 + b2 α + c2 = 0. Eliminating α, we get (c1a2c2a1)2 = (b1c2b2c1) × (a1b2a2b1). This condition can easily be remembered by cross-multiplication method as shown in the following figure.
 
62539.png
 
(Bigger cross product)2 = Product of two smaller crosses
 
The common root is given by
α = 62532.png.
 

Note: The common root can also be obtained by making the coefficient of x2 common to two given equations and then subtracting the two equations. The other roots of the given equations can be determined by using the relations between their roots and coefficients.

Condition for both roots common

Let α, β be the common roots of the quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0.
 
Then both the equations are identical, hence 62500.png = 62494.png = 62488.png.

 

Notes:
  • If two quadratic equations with real coefficients have a non-real complex common root then both the the roots will be common, i.e., both the equations will be the same. So the coefficients of the corresponding powers of x will have proportional values.
  • If two quadratic equations with rational coefficients have a common irrational root 63134.png then both roots will be common, i.e., no two different quadratic equations with rational coefficients can have a common irrational root 63128.png.




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