To solve the quadratic equation \( x^2 - 10x + 24 = 0 \), we can either factorize it or use the quadratic formula. Let's try to factor it first.
We are looking for two numbers that multiply to \( 24 \) (the constant term) and add up to \( -10 \) (the coefficient of \( x \)). The numbers that fit this criterion are \( -4 \) and \( -6 \).
Thus, we can factor the equation as follows:
\[ x^2 - 10x + 24 = (x - 4)(x - 6) = 0 \]
Setting each factor equal to zero gives us the potential solutions:
- \( x - 4 = 0 \) ⟹ \( x = 4 \)
- \( x - 6 = 0 \) ⟹ \( x = 6 \)
So, the solutions to the equation are \( x = 4 \) and \( x = 6 \).
Now let's compare this with the given options:
A. Both \( x=4 \) and \( x=6 \) are solutions - This is correct. B. Only \( x=4 \) is a solution - This is incorrect. C. Only \( x=6 \) is a solution - This is incorrect. D. Neither \( x=4 \) nor \( x=6 \) is a solution - This is incorrect.
Thus, the correct option is:
A. Both x=4 and x=6 are solutions.