1.

X square+2(a+b)x+2(a square+b square)=0 has no real roots

Answer»

Answer :

no real roots

Step-by-step explanation :

➤ Quadratic Polynomials :

     ✯ It is a polynomial of degree 2

     ✯ General form :

               ax² + bx + C  = 0

               \boxed{\bf x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}

       

     ✯ Determinant, D = b² - 4ac

     ✯ Based on the value of Determinant, we can define the nature of roots.

             D > 0 ; real and unequal roots

             D = 0 ; real and equal roots

             D < 0 ; no real roots i.e., imaginary

     ✯ RELATIONSHIP between zeroes and coefficients :

               ✩ Sum of zeroes = -b/a

               ✩ Product of zeroes = c/a

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Given polynomial,

x² + 2(a+b)x + 2(a²+b²) = 0

It is of the form ax² + bx + c = 0

  • a = 1
  • b = 2(a+b)
  • c = 2(a²+b²)

Discriminant, D = b² - 4ac

                           = [2(a+b)]² - 4(1)[2(a²+b²)]

                           = 4(a+b)² - 8(a² + b²)

                           = 4(a² + b² + 2AB) - 8(a² + b²)

                           = 4a² + 4b² + 8ab - 8a² - 8b²

                           = -4a² - 4b² + 8ab

                           = -4(a² + b² - 2ab)

                           = -4(a - b)²

 ∵ D < 0 ,it has no real roots


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