UGCNET Questions

The equation (x − 3)(x + 5) = 0 is satisfied when either factor is zero. Setting x − 3 = 0 gives x = 3, and setting x + 5 = 0 gives x = −5. The sum of the roots is therefore 3 + (−5) = −2. This also agrees with the general property that for x² + bx + c = 0, the sum of the roots is −b, which is consistent if the equation is expanded.

Option A:
Option A correctly adds the two root values, 3 and −5, to obtain −2. It captures the essence of the relationship between factorized form and the numerical roots.

Option B:
Option B, 2, would arise if one mistakenly subtracted −5 from 3 or ignored the sign of one root, but that does not reflect the actual arithmetic.

Option C:
Option C, −8, is the product of the roots, not their sum. It reflects a different relation in quadratics and confuses two separate concepts.

Option D:
Option D, 8, would require both roots to be positive or both negative with different magnitudes, which is not the case for this factorization.

Equal real roots occur when the discriminant of a quadratic equation is zero. For the equation 2x² − kx + 8 = 0, the discriminant is b² − 4ac, which becomes k² − 4(2)(8). Setting this equal to zero gives k² − 64 = 0. Solving, we get k² = 64 and hence k = ±8, but the given options only include 8 as a valid choice. Therefore, k = 8 satisfies the equal roots condition using the standard discriminant rule.

Option A:
Option A, 4√2, when squared gives 32, and substituting into the discriminant k² − 64 yields 32 − 64 = −32, which is negative. A negative discriminant would produce complex, not equal real, roots.

Option B:
Option B leads to k² = 64, so the discriminant becomes 64 − 64 = 0. This matches the precise condition for equal real roots in a quadratic and is consistent with the formula b² − 4ac = 0.

Option C:
Option C, 16, gives k² = 256 and a discriminant of 256 − 64 = 192, which is positive and corresponds to two distinct real roots rather than equal ones.

Option D:
Option D, 16√2, squares to 512, making the discriminant 512 − 64 = 448. This again yields unequal real roots, not the required equal roots.