The equation ${X^2} - 6X + 8 + \lambda \left( {{X^2} - 4X + 3} \right) = 0,\forall $ $\lambda \in R - \left{ { -1} \right}$ has
- $\left( A \right)$ real and unequal roots & for all $\lambda $
- $\left( B \right)$ real root for $\lambda < 0$
- $\left( C \right)$ real root for $\lambda > 0$
- $\left( D \right)$ real and unequal roots for $\lambda = 0$
The equation is (1+lambda)X^2 - (6+4lambda)X + (8+3lambda) = 0. For real and unequal roots, the discriminant D > 0. D = (6+4lambda)^2 - 4(1+lambda)(8+3lambda) = 36 + 48lambda + 16lambda^2 - 4(8 + 3lambda + 8lambda + 3lambda^2) = 16lambda^2 + 48lambda + 36 - 12lambda^2 - 44lambda - 32 = 4lambda^2 + 4lambda + 4. Since 4(lambda^2 + lambda + 1) is always positive for all real lambda, the roots are always real and unequal.
The given equation x squared minus 6 x plus 8 plus lambda times the quantity x squared minus 4 x plus 3 equals 0 represents a family of curves passing through the intersection points of x squared minus 6 x plus 8 equals 0 and x squared minus 4 x plus 3 equals 0. Factoring the first equation gives roots at x equals 2 and x equals 4, while factoring the second gives roots at x equals 1 and x equals 3. Since there are no common roots, the family of equations has two distinct real roots for any real value of lambda. Therefore, the equation has real and unequal roots for all lambda.