To solve the equation $x^3 + 24x^2 + 44x = 0$, let's start by factoring the expression.

Given:

$x^3 + 24x^2 + 44x = 0$

Step 1: Factor out the common term $x$.

$x(x^2 + 24x + 44) = 0$

Now, we have two factors:

1. $x = 0$
2. $x^2 + 24x + 44 = 0$

Step 2: Solve the quadratic equation $x^2 + 24x + 44 = 0$.

We can solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For $ax^2 + bx + c = 0$, the coefficients are:

• $a = 1$
• $b = 24$
• $c = 44$

Substituting these into the formula:

$x = \frac{-24 \pm \sqrt{24^2 - 4(1)(44)}}{2(1)}$

$x = \frac{-24 \pm \sqrt{576 - 176}}{2}$

$x = \frac{-24 \pm \sqrt{400}}{2}$

$x = \frac{-24 \pm 20}{2}$

Step 3: Find the roots.

1. $x = \frac{-24 + 20}{2} = \frac{-4}{2} = -2$
2. $x = \frac{-24 - 20}{2} = \frac{-44}{2} = -22$

Final Solutions:

The roots of the equation are:

$x = 0, \, x = -2, \, \text{and} \, x = -22$

Summary:

The solutions to the equation $x^3 + 24x^2 + 44x = 0$ are $x = 0$, $x = -2$, and $x = -22$.

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Would you like more details or have any questions?

Here are 5 related questions:

1. How do you factor a cubic equation with more complex coefficients?
2. What is the significance of the discriminant in the quadratic formula?
3. How can you solve a cubic equation using the Rational Root Theorem?
4. What are some graphical interpretations of the roots of polynomial equations?
5. How can you verify solutions to a polynomial equation by substitution?

Tip: Always check for a common factor first when solving polynomial equations!