Solve the Cubic Equation: x³ = x² + 2x Step-by-Step

Q: Why can't I just divide both sides by x to simplify?

When you divide by x, you're assuming x ≠ 0. But what if x = 0 is actually a solution? You'd lose it completely! Always factor instead of dividing by variables.

Q: How do I factor x² - x - 2 quickly?

Look for two numbers that multiply to -2 and add to -1. Try: -2 × 1 = -2, and -2 + 1 = -1. Perfect! So $ x^2 - x - 2 = (x-2)(x+1) $.

Q: Do I need to check all three solutions?

Yes! Always verify each solution by substituting back into the original equation. This catches any algebra mistakes and confirms your factoring was correct.

Q: What if the quadratic doesn't factor nicely?

Use the quadratic formula: $ x = \frac{-b ± \sqrt{b^2-4ac}}{2a} $. But first, always try factoring since it's usually faster when it works!

Q: Can cubic equations have more than 3 solutions?

No! A cubic equation can have at most 3 real solutions. This follows from the fundamental theorem of algebra - a polynomial of degree n has exactly n roots (counting multiplicity).

Cubic Factoring with Zero Product Property

$x^3=x^2+2x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's start by finding the value of X.

00:09 First, arrange the equation so that, the right side equals zero.

00:16 Now, factor the equation into terms with X squared.

00:29 Next, take out the common factor from the parentheses.

00:38 We’re looking for solutions, that make each factor zero.

00:42 This gives us one solution.

00:45 Now, let's find the second solution.

00:48 Use the trinomial method; look at the coefficients.

00:52 Find two numbers whose sum is B, or negative one.

00:57 And their product is C, or negative two.

01:01 These numbers fit! Let's put them into the multiplication.

01:08 Again, find which solution, zeros each factor.

01:21 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^3=x^2+2x$

Step-by-step solution

To solve the problem $x^3 = x^2 + 2x$ , follow these steps:

Step 1: Re-arrange the equation to have all terms on one side:
$x^3 - x^2 - 2x = 0$ .
Step 2: Factor out the greatest common factor (GCF), which is $x$ :
$x(x^2 - x - 2) = 0$ .
Step 3: Factor the quadratic expression $x^2 - x - 2$ :
The factors of $-2$ that add up to $-1$ are $-2$ and $1$ . Thus, $x^2 - x - 2 = (x-2)(x+1)$ .
Step 4: Combine the factored terms:
$x(x-2)(x+1) = 0$ .

Each factor can be set to zero to find the solutions:

$x = 0$ .
$x - 2 = 0$ , so $x = 2$ .
$x + 1 = 0$ , so $x = -1$ .

The solutions to the equation are $x = 0, -1, 2$ .

Therefore, the correct choice from the given options is:
$x = 0, -1, 2$ .

Final Answer

$x=0,-1,2$

Key Points to Remember

Essential concepts to master this topic

Rule: Move all terms to one side before factoring
Technique: Factor out GCF x first: x(x² - x - 2) = x(x-2)(x+1)
Check: Verify each solution: 0³ = 0² + 2(0) = 0 ✓

Common Mistakes

Avoid these frequent errors

Dividing both sides by x immediately
Don't divide both sides by x at the start = you'll lose the solution x = 0! This eliminates a valid root from your answer set. Always rearrange to standard form and factor out x instead.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

Why can't I just divide both sides by x to simplify?

When you divide by x, you're assuming x ≠ 0. But what if x = 0 is actually a solution? You'd lose it completely! Always factor instead of dividing by variables.

How do I factor x² - x - 2 quickly?

Look for two numbers that multiply to -2 and add to -1. Try: -2 × 1 = -2, and -2 + 1 = -1. Perfect! So $x^2 - x - 2 = (x-2)(x+1)$ .

Do I need to check all three solutions?

Yes! Always verify each solution by substituting back into the original equation. This catches any algebra mistakes and confirms your factoring was correct.

What if the quadratic doesn't factor nicely?

Use the quadratic formula: $x = \frac{-b ± \sqrt{b^2-4ac}}{2a}$ . But first, always try factoring since it's usually faster when it works!

Can cubic equations have more than 3 solutions?

No! A cubic equation can have at most 3 real solutions. This follows from the fundamental theorem of algebra - a polynomial of degree n has exactly n roots (counting multiplicity).

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Solve the Cubic Equation: x³ = x² + 2x Step-by-Step

Cubic Factoring with Zero Product Property

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just divide both sides by x to simplify?

How do I factor x² - x - 2 quickly?

Do I need to check all three solutions?

What if the quadratic doesn't factor nicely?

Can cubic equations have more than 3 solutions?

🌟 Unlock Your Math Potential

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