Roots of a Quadratic Equation

The governing concept is the root condition: if a number is a root of a polynomial equation, substituting that number makes the entire expression equal to zero. Put x = −4 into 2x² + px − 8 = 0: 2(−4)² + p(−4) − 8 = 0. Since (−4)² = 16, we obtain 32 − 4p − 8 = 0, so 24 − 4p = 0. Therefore 4p = 24 and p = 6. Hence option A is correct. A common error is to treat p(−4) as +4p; that sign mistake can suggest a negative value such as option B. Options C and D do not satisfy the equation when checked by substitution.

The governing rule is that a stated root must make the quadratic expression equal to zero. Substitute x = 2 into x² + kx − 14 = 0: 2² + k(2) − 14 = 0. This becomes 4 + 2k − 14 = 0, so 2k − 10 = 0. Hence 2k = 10 and k = 5. Therefore option A is correct. The constant term remains −14; it is not replaced or sign-changed during substitution. Option B would arise from an incorrect sign while transposing terms. Options C and D can be rejected directly because substituting k = 7 or k = −7 gives a non-zero value instead of zero. A final check with k = 5 gives 4 + 10 − 14 = 0.

The governing concept is the multiplicative inverse, or reciprocal. For any non-zero number n, its reciprocal is 1/n, and multiplying the two gives n × 1/n = 1. In this case, 4 × 1/4 = 1, so 1/4 is the reciprocal of 4. Therefore option A is correct. The roots are not equal because 4 and 1/4 have different values. Their sum is 4 + 1/4 = 17/4, not 1, so option C is false. Their product is 1 rather than 0, which rules out option D. The conclusion follows from the values of the roots themselves; there is no need to reconstruct or solve the quadratic equation.

A reciprocal is the multiplicative inverse of a non-zero number. For a number n, its reciprocal is 1/n, and n × 1/n = 1. The two given roots satisfy 5 × 1/5 = 1, so 1/5 is the reciprocal of 5. Therefore option A is correct. They are not equal because 5 is different from 1/5. Their sum is 5 + 1/5 = 26/5, not 1, eliminating option C. Their product is 1, not 0, eliminating option D. This question tests the relationship between the roots rather than requiring the quadratic equation to be solved or reconstructed. The product calculation is sufficient to establish the reciprocal relationship.

Factor out the common factor x from the equation: x² − 16x = x(x − 16) = 0. The zero-product property states that if a product is zero, at least one factor must be zero. Therefore x = 0 or x − 16 = 0. Solving the second factor gives x = 16, so the two roots are 0 and 16. Hence option A is correct. A common error is to read the coefficient −16 as a root and write −16; however, the factor is x − 16, whose zero is positive 16. Options C and D introduce values not obtained from either factor. Substitution confirms the result: both 0² − 16(0) and 16² − 16(16) equal zero.

The governing principle is that dividing an equation by a variable factor can discard a solution when that factor itself may be zero. The original equation factors as x² − 5x = x(x − 5) = 0. By the zero-product property, either x = 0 or x − 5 = 0, which gives x = 5. If we divide by x immediately, we implicitly assume x is not zero and obtain only x − 5 = 0. Thus the root x = 0 is lost. Option A is correct. The root x = 5 is retained, while −5 and 1 do not satisfy the original equation. The safe method is to factor first and consider every factor separately.