Nature of Roots of Quadratic Polynomial


 
 
Concept Explanation
 

Nature of Roots of Quadratic Polynomial

Nature of Roots of Quadratic Polynomial:

The Value of Discriminant   large D=b^2-4ac decides the nature of roots of a quadratic Polynomial.

Case 1: If large D = 0, then the roots are real and equal

Case 2: If large D>0, then the roots are real and unequal.

Case 3: If large D< 0, then the roots are imaginary

Example: Without solving the equation identify the nature of roots

large 2x^2+5x+5=0

Solution: Comparing the equation with the standard form

a = 2, b = 5 and c = 5, Then

large D=b^2-4ac

      large =5^2-4 ;X;2;X;5=25-40=-15

As the discriminant is less than zero so the roots are imaginary

Example: Find the value of k for which the equation has real and equal roots

large x^2+7(3+2k)-2x(1+3k)=0

Solution: Comparing the equation with the standard form

a = 1, b =  -2(1+3k)  and c = 7(3+2k), Then

large D=b^2-4ac

large =(-2(1+3k))^2-4;X;1;X;(7(3+2k))

large = 4(1+3k)^2-28(3+2k)

large = 4[(1+9k^2+6k)-7(3+2k)]

large = 4[1+9k^2+6k-21-14k]

large = 4[9k^2-8k-20]

large = 4[9k^2-18k+10k-20]

large = 4[9k(k-2)+10(k-2)]

large = 4[(9k+10)(k-2)]

As the roots are real and equal, so D = 0

large 4[(9k+10)(k-2)]=0

large either; k= -frac{10}{9} ;or; k =2

Sample Questions
(More Questions for each concept available in Login)
Question : 1

The number of real roots of equation:3^{2x^2-7x+7}=9 is:

Right Option : C
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Explanation
Question : 2

Find the nature of the roots of  frac{2}{7}x^2+5x+42=0

Right Option : B
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Explanation
Question : 3

Find the nature of the roots of  2x^2+5x+6=0

Right Option : B
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Explanation
 
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