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Quadratic Equations

If a0, a1, a2, …, an are real numbers and 'n' is a non-negative integer, then a function p(x) = (a0 + a1x + a2x2 + a3 x3 +…+ an xn) is called a polynomial in x over reals. The value of 'n' in a given polynomial is called the degree of the polynomial.

The most general form of quadratic equation is ax2 + bx + c = 0 where a, b and c are real constants, a 0 and x is a real variable.
 

Example : 3x2 + 5x + 9 = 0 , x2 + 9 = 0, 2x2 + 5x + 7= 0, x2 - x + 2 = 0 and = 0

 

Roots of a Quadratic Equation
If the two numbers α and β are the two roots of the quadratic equation, p(x) = ax2 + bx + c = 0, then α and β are called the roots of the quadratic equation p(x).
For x2 - x - 6 = 0, the roots are 3 and (-2)

 

 
Note
If ax2 + bx + c becomes 0 for x = α then α is known as the root of the quadratic equation

 

Example

Which of the following are quadratic equations?
(a) 2x2 + 5x + 3 = 0                     (b) 7x + 5 = 0
(c)                       (d) x3 + 1 = 0
(e) 2x2 + 5x + 3 = 2x2 + 4x + 5.

Solution

(a) Quadratic equation
(b) Linear equation
(c) Quadratic equation
(d) It is not a quadratic equation.
(e) Transforming R. H. S. to L. H. S., we get
2x2 + 5x + 3 - 2x2 - 4x - 5 = 0

x - 2 = 0
 ...  It is a linear equation.

 
Example

For the quadratic equation find out which of the following are solutions
(a) x =         (b) x =       (c) x = .

Solution

(a) Substituting x = on L.H.S. of the equation we get,
     
      L.H.S. ≠ R.H.S.
 Therefore, x = is not a solution.
(b) Substituting x = on the L.H.S.,
       

         = = R.H.S.

Therefore, x = is a solution of the equation.
(c) Substituting x = on the L.H.S. we get,
       
         =     = R.H.S.
Therefore, x= - is also a solution of the equation.

 




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