If c is a constant and the equation x² - 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.
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 this yields ( 16 - 4c < 0 ), leading to ( c > 4 ).
Choosing ( c = -6 ) yields a discriminant of ( 16 - 4(-6) = 16 + 24 = 40 ), which is greater than zero. This means the equation has two distinct real roots, contradicting the requirement for no real roots.
If ( c = -4 ), the discriminant becomes ( 16 - 4(-4) = 16 + 16 = 32 ). This value is also positive, indicating that the quadratic has two real roots, thus failing to satisfy the condition of having no real roots.
Setting ( c = 2 ) results in a discriminant of ( 16 - 4(2) = 16 - 8 = 8 ). Since this value is positive, it confirms the presence of two real roots, which does not meet the requirement for the quadratic to lack real roots.
If ( c = 4 ), the discriminant is ( 16 - 4(4) = 16 - 16 = 0 ). A discriminant of zero signifies that there is exactly one real root (a repeated root), not the absence of real roots. This choice is also incorrect.
When ( c = 6 ), the discriminant equals ( 16 - 4(6) = 16 - 24 = -8 ). A negative discriminant indicates that the equation has no real roots, fulfilling the condition set by the question.
For the quadratic equation ( x^2 - 4x + c = 0 ) to lack real roots, the constant ( c ) must exceed 4. Among the provided options, only ( c = 6 ) results in a negative discriminant, confirming that it is the only value that allows for no real roots in the equation. Thus, ( c = 6 ) is the only suitable choice.
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