To solve the equation x² – 2x – 24 = 0, we can start by factoring the quadratic expression. We are looking for two numbers that multiply to -24 (the constant term) and add up to -2 (the coefficient of the x term).
The numbers that satisfy these conditions are -6 and +4 because:
- -6 × 4 = -24
- -6 + 4 = -2
So, we can factor the equation as:
(x – 6)(x + 4) = 0
Next, we can set each factor equal to zero to find the values of x:
- x – 6 = 0 → x = 6
- x + 4 = 0 → x = -4
Therefore, the solutions to the equation x² – 2x – 24 = 0 are:
- x = 6
- x = -4
To verify, we can substitute each value back into the original equation:
- For x = 6:
6² – 2(6) – 24 = 36 – 12 – 24 = 0 - For x = -4:
(-4)² – 2(-4) – 24 = 16 + 8 – 24 = 0
Both values satisfy the equation, confirming that the solutions are correct.