Question

1. solve the equation $x^3 - 3x^2 + 25x - 75 = 0$ in the complex number system.

\bigcirc $x = 2i, x = -2i, x = 25$
\bigcirc $x = 5i, x = -5i, x = 3$
\bigcirc $x = 3i, x = -3i, x = 5$
\bigcirc $x = 25i, x = -25i, x = 2$

Explanation:

Step 1: Factor by grouping

We start with the equation \( x^3 - 3x^2 + 25x - 75 = 0 \). Group the first two terms and the last two terms:

$$ (x^3 - 3x^2) + (25x - 75) = 0 $$

Factor out the greatest common factor from each group:

$$ x^2(x - 3) + 25(x - 3) = 0 $$

Now, we can factor out \( (x - 3) \) from both terms:

$$ (x - 3)(x^2 + 25) = 0 $$

Step 2: Solve for \( x \)

Set each factor equal to zero and solve for \( x \).

For \( x - 3 = 0 \):

$$ x = 3 $$

For \( x^2 + 25 = 0 \):

Subtract 25 from both sides:

$$ x^2 = -25 $$

Take the square root of both sides. Remember that \( \sqrt{-1} = i \), so:

$$ x = \pm \sqrt{-25} = \pm 5i $$

So the solutions to the equation are \( x = 3 \), \( x = 5i \), and \( x = -5i \), which corresponds to option B.

Answer:

B. \( x = 5i, x = -5i, x = 3 \)