Solve the Cubic Equation: x³ - 5x² = -6x Step-by-Step

Q: Why can't I just divide both sides by x to make it simpler?

Because x could equal zero! When you divide by a variable, you're assuming it's not zero, which makes you lose solutions. In this problem, x = 0 is actually one of the correct answers.

Q: How do I know when to factor out x?

Look for every term containing x. In $ x^3 - 5x^2 + 6x = 0 $, all three terms have x, so you can factor it out as the greatest common factor.

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

If $ x^2 - 5x + 6 $ didn't factor easily, you could use the quadratic formula. But always check if it factors first - it's usually faster!

Q: How do I check if my factoring is correct?

Expand it back out! Multiply $ x(x-2)(x-3) $ and see if you get the original equation. This catches factoring mistakes before you solve.

Q: Why do I get three solutions for a cubic equation?

A cubic equation (degree 3) can have up to 3 real solutions. The degree tells you the maximum number of solutions possible. Sometimes you might get repeated solutions too.

Cubic Equations with Factoring Method

$x^3-5x^2=-6x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Arrange the equation so that one side equals 0

00:12 Factor X³ into X² and X

00:17 Factor X² into X and X

00:20 Find the common factor

00:27 Take out the common factor from the parentheses

00:38 Find what makes each factor equal zero

00:42 This is one solution

00:45 Now find what makes the parentheses equal zero

00:53 Let's examine the trinomial coefficients

00:56 We want to find 2 numbers that sum to (-5)

01:00 and their product is (6)

01:05 These are the matching numbers, with them we'll build the trinomial

01:12 Find what makes each factor equal zero

01:16 This is the second solution

01:21 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^3-5x^2=-6x$

Step-by-step solution

To solve the equation $x^3 - 5x^2 = -6x$ , we'll use factoring:

Step 1: Start by rearranging the equation: $x^3 - 5x^2 + 6x = 0$ .
Step 2: Factor out the greatest common factor, which is $x$ :

$x(x^2 - 5x + 6) = 0$

Step 3: The quadratic $x^2 - 5x + 6$ can be factored further:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

Step 4: Substitute back into the equation:

$x(x - 2)(x - 3) = 0$

Step 5: Use the zero-product property to find the solutions:

Set each factor to zero: $x = 0$ , $x - 2 = 0$ , and $x - 3 = 0$ .

Step 6: Solve each equation:

$x = 0$

$x = 2$

$x = 3$

Therefore, the solutions to the equation $x^3 - 5x^2 = -6x$ are $x = 0, 2, 3$ .

Final Answer

$x=0,2,3$

Key Points to Remember

Essential concepts to master this topic

Zero Product Property: If ab = 0, then a = 0 or b = 0
Factoring Strategy: Factor out x first: $x(x^2 - 5x + 6) = 0$
Verification: Check each solution: $2^3 - 5(2^2) = 8 - 20 = -12 = -6(2)$ ✓

Common Mistakes

Avoid these frequent errors

Dividing both sides by x to eliminate it
Don't divide both sides by x to get $x^2 - 5x = -6$ = you lose the solution x = 0! Dividing by a variable that could be zero eliminates valid solutions. Always move all terms to one side and factor out x instead.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

FAQ

Everything you need to know about this question

Why can't I just divide both sides by x to make it simpler?

Because x could equal zero! When you divide by a variable, you're assuming it's not zero, which makes you lose solutions. In this problem, x = 0 is actually one of the correct answers.

How do I know when to factor out x?

Look for every term containing x. In $x^3 - 5x^2 + 6x = 0$ , all three terms have x, so you can factor it out as the greatest common factor.

What if the quadratic doesn't factor nicely?

If $x^2 - 5x + 6$ didn't factor easily, you could use the quadratic formula. But always check if it factors first - it's usually faster!

How do I check if my factoring is correct?

Expand it back out! Multiply $x(x-2)(x-3)$ and see if you get the original equation. This catches factoring mistakes before you solve.

Why do I get three solutions for a cubic equation?

A cubic equation (degree 3) can have up to 3 real solutions. The degree tells you the maximum number of solutions possible. Sometimes you might get repeated solutions too.

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Solve the Cubic Equation: x³ - 5x² = -6x Step-by-Step

Cubic Equations with Factoring Method

❤️ 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 make it simpler?

How do I know when to factor out x?

What if the quadratic doesn't factor nicely?

How do I check if my factoring is correct?

Why do I get three solutions for a cubic equation?

🌟 Unlock Your Math Potential

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