If c is a constant and the equation x ^ 2 - 4x + c = 0 has no real roots, which of the following could be the value of c?

Rationale

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

For the quadratic equation (x^2 - 4x + c = 0) to have no real roots, the discriminant must be less than zero. The discriminant is calculated as (b^2 - 4ac), which in this case translates to ((-4)^2 - 4(1)(c) < 0). Simplifying gives (16 - 4c < 0), leading to (c > 4).

A (-6) YOUR ANSWER

If (c = -6), the discriminant becomes (16 - 4(-6) = 16 + 24 = 40), which is greater than zero. This indicates that there are two distinct real roots for the equation, contradicting the condition that it has no real roots.

B (-4) YOUR ANSWER

Choosing (c = -4) results in a discriminant of (16 - 4(-4) = 16 + 16 = 32), also greater than zero. Thus, this value of (c) similarly leads to two real roots, failing to satisfy the requirement of having no real roots.

C (2) YOUR ANSWER

For (c = 2), the discriminant calculates to (16 - 4(2) = 16 - 8 = 8), which is still positive. This means that the equation has two real roots, thus not fulfilling the condition of no real roots.

D (4) YOUR ANSWER

Setting (c = 4) gives a discriminant of (16 - 4(4) = 16 - 16 = 0). A discriminant of zero indicates that the equation has exactly one real root (a repeated root), which again does not satisfy the condition of having no real roots.

E (6) CORRECT

When (c = 6), the discriminant is (16 - 4(6) = 16 - 24 = -8), which is less than zero. This indicates that the equation indeed has no real roots, satisfying the condition perfectly.

Conclusion

For the quadratic equation (x^2 - 4x + c = 0) to lack real roots, (c) must exceed 4, ensuring a negative discriminant. Among the provided choices, only (c = 6) meets this requirement, demonstrating the importance of the discriminant in determining the nature of roots in quadratic equations.

If c is a constant and the equation x ^ 2 - 4x + c = 0 has no real roots, which of the following could be the value of c?

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

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