https://file.aiquickdraw.com/mathbot/file/1729566277622vxzrdxzq.jpg

The equation from the image is:

\[
-x^3 + 5x^2 + 25x - 125 = 0
\]

### Step 1: Simplify the equation by factoring out \(-1\):

\[
x^3 - 5x^2 - 25x + 125 = 0
\]

### Step 2: Use factoring by grouping.

Group the terms in pairs:

\[
(x^3 - 5x^2) - (25x - 125) = 0
\]

Factor out the common terms from each group:

\[
x^2(x - 5) - 25(x - 5) = 0
\]

Now factor out the common binomial \((x - 5)\):

\[
(x - 5)(x^2 - 25) = 0
\]

### Step 3: Factor the remaining quadratic.

Notice that \(x^2 - 25\) is a difference of squares:

\[
x^2 - 25 = (x - 5)(x + 5)
\]

Thus, the fully factored form is:

\[
(x - 5)(x - 5)(x + 5) = 0
\]

### Step 4: Solve for \(x\).

Set each factor equal to zero:

\[
x - 5 = 0 \quad \text{or} \quad x + 5 = 0
\]

This gives the solutions:

\[
x = 5 \quad \text{(with multiplicity 2)} \quad \text{or} \quad x = -5
\]

### Final Answer:

The solutions are \(x = 5\) (with multiplicity 2) and \(x = -5\).

Would you like further details or have any questions about this process?

Here are some related questions for practice:
1. How would you factor a cubic equation with no obvious grouping?
2. What is the significance of the multiplicity of a root?
3. How can you solve a cubic equation that doesn’t factor easily?
4. How do you recognize and factor a difference of squares?
5. How can you verify the solutions to a polynomial equation?

**Tip:** Always check if the equation can be simplified or factored before using more advanced methods like synthetic division or the Rational Root Theorem.

Solve algebraically for all values of x: -x^3 + 5x^2 + 25x - 125 = 0

Learn how to solve the cubic equation -x^3 + 5x^2 + 25x - 125 = 0 with a step-by-step factoring approach. This solution covers key algebraic techniques, including factoring by grouping and recognizing differences of squares. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation