To solve the equation \( x^3 - 2x^2 - 25x = -50 \), we first rearrange it into standard form:
\[ x^3 - 2x^2 - 25x + 50 = 0 \]
Next, we will factor the polynomial \( x^3 - 2x^2 - 25x + 50 \). We can use the Rational Root Theorem to find potential rational roots, which could be factors of the constant term (50). The possible rational roots could be \( \pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50 \).
Let's test these values by substituting them into the polynomial.
- Testing \( x = 2 \):
\[ 2^3 - 2(2^2) - 25(2) + 50 = 8 - 8 - 50 + 50 = 0 \]
Since \( x = 2 \) is a root, we can use synthetic division to factor \( x - 2 \) out of the polynomial.
Using synthetic division with \( x = 2 \):
2 | 1 -2 -25 50
| 2 0 -50
-----------------------
1 0 -25 0
The quotient is \( x^2 - 25 \).
Thus, we can factor the polynomial as:
\[ (x - 2)(x^2 - 25) = 0 \]
Next, we notice that \( x^2 - 25 \) is a difference of squares, which can be factored further:
\[ x^2 - 25 = (x - 5)(x + 5) \]
Now we combine all the factors:
\[ (x - 2)(x - 5)(x + 5) = 0 \]
Setting each factor to zero gives the solutions:
- \( x - 2 = 0 \) → \( x = 2 \)
- \( x - 5 = 0 \) → \( x = 5 \)
- \( x + 5 = 0 \) → \( x = -5 \)
Thus, the solutions to the equation \( x^3 - 2x^2 - 25x + 50 = 0 \) are:
\[ \boxed{2, 5, -5} \]
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