Question 15

The value of c must be greater than 4 for the equation to have no real roots.

For the equation x² - 4x + c = 0 to have no real roots, the discriminant must be less than zero. The discriminant is calculated as b² - 4ac, which in this case is (-4)² - 4(1)(c) = 16 - 4c. Setting this less than zero gives us 16 - 4c < 0, leading to c > 4. Therefore, the possible value for c that fits this criterion is 6.

A) -6

This value is incorrect because substituting c = -6 into the discriminant gives 16 - 4(-6) = 16 + 24 = 40, which is greater than zero, indicating that there are two real roots.

B) -4

This value is also incorrect. If c = -4, then the discriminant becomes 16 - 4(-4) = 16 + 16 = 32, which is still greater than zero, resulting in two real roots.

C) 2

This option is incorrect as well. For c = 2, the discriminant would be 16 - 4(2) = 16 - 8 = 8, which is greater than zero, meaning the equation has two real roots.

D) 4

This value is incorrect. With c = 4, the discriminant results in 16 - 4(4) = 16 - 16 = 0, which indicates that there is exactly one real root, not no real roots.

E) 6

This is the correct option, as substituting c = 6 results in the discriminant being 16 - 4(6) = 16 - 24 = -8, which is less than zero, confirming that the equation has no real roots.

Conclusion

The only value for c that ensures the quadratic equation has no real roots is 6, as it satisfies the condition derived from the discriminant being negative. All other options result in either a positive discriminant, indicating two real roots, or a zero discriminant, indicating one real root. Thus, option E is definitively the correct answer.