Find the solution(s) to the quadratic equation x2−10x+24

Question

, and then select the correct answer from the following options.(1 point)
Responses

Only x=6 is a solution
Only x=6 is a solution

Both x=4 and x=6 are solutions
Both x=4 and x=6 are solutions

Neither x=4 nor x=6 is a solution.
Neither x=4 nor x=6 is a solution.

Only x=4 is a solution

Answer 1

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.