Question 9

The value of c that ensures the equation has no real roots is 6.

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 \( b^2 - 4ac \). In this case, with \( a = 1 \), \( b = -4 \), and \( c = c \), the discriminant becomes \( (-4)^2 - 4(1)(c) = 16 - 4c \). Setting this less than zero gives \( 16 - 4c < 0 \), which simplifies to \( c > 4 \). Therefore, the value of \( c \) could be 6.

A) -6

Choosing -6 would result in a discriminant of \( 16 - 4(-6) = 16 + 24 = 40 \), which is greater than zero. Hence, this value of \( c \) would yield two real roots, contradicting the condition of having no real roots.

B) -4

If we select -4, the discriminant would be \( 16 - 4(-4) = 16 + 16 = 32 \). This value is also greater than zero, meaning the equation would have two real roots, which does not satisfy the requirement of having no real roots.

C) 2

Choosing 2 gives a discriminant of \( 16 - 4(2) = 16 - 8 = 8 \), which is again greater than zero. This indicates that the equation would have two real roots, thus failing to meet the criteria for \( c \).

D) 4

If \( c \) is set to 4, the discriminant becomes \( 16 - 4(4) = 16 - 16 = 0 \). A discriminant of zero indicates there is exactly one real root, not no real roots, so this value does not satisfy the condition.

E) 6

With \( c = 6 \), the discriminant is calculated as \( 16 - 4(6) = 16 - 24 = -8 \), which is less than zero. This confirms that the equation \( x^2 - 4x + 6 = 0 \) has no real roots.

Conclusion

The only value of \( c \) that ensures the quadratic equation has no real roots is 6, as it results in a negative discriminant. All other options either yield a positive or zero discriminant, which indicates the presence of real roots. Thus, 6 is the definitive choice for the given condition.