To solve the quadratic equation \( x^2 - 10x + 24 = 0 \), we can factor the expression.
We are looking for two numbers that multiply to \( 24 \) (the constant term) and add to \( -10 \) (the coefficient of the linear term).
The numbers \( -4 \) and \( -6 \) meet these criteria:
\[
(-4) \times (-6) = 24
\]
\[
(-4) + (-6) = -10
\]
So we can factor the quadratic as:
\[
(x - 4)(x - 6) = 0
\]
Setting each factor equal to zero gives us:
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
Thus, the solutions to the equation \( x^2 - 10x + 24 = 0 \) are \( x = 4 \) and \( x = 6 \).
Among the provided options, the correct answer is:
**Both x=4 and x=6 are solutions.**