Solve the Cubic Equation: 9x³ - 12x² = 0

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

Because x could equal zero! When you divide by x, you're assuming x ≠ 0, which eliminates the solution x = 0. Always factor first to preserve all solutions.

Q: How do I find the greatest common factor of polynomial terms?

Look at both the coefficients and the variables. For $ 9x^3 $ and $ 12x^2 $: GCF of 9 and 12 is 3, GCF of $ x^3 $ and $ x^2 $ is $ x^2 $, so the GCF is $ 3x^2 $.

Q: What is the zero-product property exactly?

It states that if the product of factors equals zero, then at least one factor must be zero. So from $ 3x^2(3x - 4) = 0 $, either $ 3x^2 = 0 $ or $ 3x - 4 = 0 $.

Q: Why does x = 0 appear twice as a root?

Actually, x = 0 is a repeated root with multiplicity 2 because the factor $ 3x^2 $ contains $ x^2 $. This means the graph touches the x-axis at x = 0 but doesn't cross it.

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

Expand your factored form back out! $ 3x^2(3x - 4) = 9x^3 - 12x^2 $. If you get the original expression, your factoring is correct.

Cubic Equations with Factoring Method

Find the value of the parameter x.

$9x^3-12x^2=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Factor into components

00:16 Extract common factor

00:32 Find what zeros each factor

00:38 Isolate the unknown, this is one solution, now let's find the second

00:55 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the value of the parameter x.

$9x^3-12x^2=0$

Step-by-step solution

To solve this problem, we will factor the given polynomial expression:

Step 1: Identify the greatest common factor (GCF) in the equation $9x^3 - 12x^2 = 0$ . The GCF of the terms $9x^3$ and $12x^2$ is $3x^2$ .

Step 2: Factor out the GCF from the polynomial:

$9x^3 - 12x^2 = 3x^2(3x - 4) = 0$ .

Step 3: Apply the zero-product property. Set each factor equal to zero:

$3x^2 = 0$
$3x - 4 = 0$

Step 4: Solve each equation for $x$ :

For $3x^2 = 0$ , divide by 3:

$x^2 = 0$ → $x = 0$ .

For $3x - 4 = 0$ , add 4 to both sides and then divide by 3:

$3x = 4$
$x = \frac{4}{3}$ .

Thus, the solutions to the equation $9x^3 - 12x^2 = 0$ are $x = 0$ and $x = \frac{4}{3}$ .

Therefore, the correct answer is:

$x=0, x=\frac{4}{3}$

Final Answer

$x=0,x=\frac{4}{3}$

Key Points to Remember

Essential concepts to master this topic

Greatest Common Factor: Always identify and factor out the GCF first
Zero-Product Property: If $ab = 0$ , then $a = 0$ or $b = 0$
Check: Substitute both solutions back: $9(0)^3 - 12(0)^2 = 0$ and $9(\frac{4}{3})^3 - 12(\frac{4}{3})^2 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Dividing both sides by x without considering x = 0
Don't divide both sides by x from $9x^3 - 12x^2 = 0$ = you'll lose the solution x = 0! This eliminates a valid root because you're dividing by zero. Always factor out the GCF completely before applying the zero-product property.

Practice Quiz

Test your knowledge with interactive questions

Find the value of the parameter x.

$$ x^2+x=0 $$

FAQ

Everything you need to know about this question

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

Because x could equal zero! When you divide by x, you're assuming x ≠ 0, which eliminates the solution x = 0. Always factor first to preserve all solutions.

How do I find the greatest common factor of polynomial terms?

Look at both the coefficients and the variables. For $9x^3$ and $12x^2$ : GCF of 9 and 12 is 3, GCF of $x^3$ and $x^2$ is $x^2$ , so the GCF is $3x^2$ .

What is the zero-product property exactly?

It states that if the product of factors equals zero, then at least one factor must be zero. So from $3x^2(3x - 4) = 0$ , either $3x^2 = 0$ or $3x - 4 = 0$ .

Why does x = 0 appear twice as a root?

Actually, x = 0 is a repeated root with multiplicity 2 because the factor $3x^2$ contains $x^2$ . This means the graph touches the x-axis at x = 0 but doesn't cross it.

How can I check if my factoring is correct?

Expand your factored form back out! $3x^2(3x - 4) = 9x^3 - 12x^2$ . If you get the original expression, your factoring is correct.

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Solve the Cubic Equation: 9x³ - 12x² = 0

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 simplify?

How do I find the greatest common factor of polynomial terms?

What is the zero-product property exactly?

Why does x = 0 appear twice as a root?

How can I check if my factoring is correct?

🌟 Unlock Your Math Potential

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