Solve the Cubic Equation: x³ + x² - 12x = 0

Cubic Equations with Factoring Method

x3+x2−12x=0 x^3+x^2-12x=0

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:05 Let's find X.
00:08 First, factor out using X.
00:19 Take out the common factor from the parentheses.
00:36 This gives us one solution that makes the equation zero.
00:42 Now let's solve the second factor.
00:49 Factor the trinomial by identifying the coefficients.
00:54 Find two numbers whose sum is B, which is 1.
00:59 And whose product is C, which is negative twelve.
01:05 These are the numbers we need. Substitute them into the parentheses.
01:10 Find what makes each factor zero.
01:15 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

x3+x2−12x=0 x^3+x^2-12x=0

2

Step-by-step solution

To solve the equation x3+x2−12x=0 x^3 + x^2 - 12x = 0 , follow these steps:

  • Step 1: Factor out the greatest common factor. The common factor here is x x .
  • Step 2: The equation becomes x(x2+x−12)=0 x(x^2 + x - 12) = 0 .
  • Step 3: Apply the zero-product property. This gives us two equations to solve: x=0 x = 0 and x2+x−12=0 x^2 + x - 12 = 0 .
  • Step 4: Solve x=0 x = 0 . This is a straightforward solution: x=0 x = 0 .
  • Step 5: Solve the quadratic equation x2+x−12=0 x^2 + x - 12 = 0 . We will factor it:
    • Factor as (x−3)(x+4)=0 (x - 3)(x + 4) = 0 .
    • Set each factor equal to zero: x−3=0 x - 3 = 0 or x+4=0 x + 4 = 0 .
    • Solving these, we obtain x=3 x = 3 and x=−4 x = -4 .

Therefore, the solutions to the equation are x=0 x = 0 , x=3 x = 3 , and x=−4 x = -4 .

Thus, the complete solution set for x x is x=0,3,−4 x = 0, 3, -4 .

3

Final Answer

x=0,3,−4 x=0,3,-4

Key Points to Remember

Essential concepts to master this topic
  • Rule: Factor out greatest common factor first before solving
  • Technique: Use zero-product property: if ab=0 ab = 0 , then a=0 a = 0 or b=0 b = 0
  • Check: Substitute each solution back: 03+02−12(0)=0 0^3 + 0^2 - 12(0) = 0 ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting the x = 0 solution
    Don't ignore the factored out x and only solve the quadratic = missing one solution! When you factor out x, you automatically get x = 0 as one solution. Always include all solutions from every factor you create.

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 do I factor out x first instead of expanding everything?

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Factoring out the greatest common factor (x) simplifies the problem! Instead of dealing with a complicated cubic, you get one easy solution (x = 0) plus a simpler quadratic to solve.

How do I know when to use the zero-product property?

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Use it whenever you have an equation in the form A×B=0 A \times B = 0 . This means at least one of the factors must equal zero, giving you separate simpler equations to solve.

What if the quadratic doesn't factor nicely?

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If x2+x−12 x^2 + x - 12 didn't factor easily, you could use the quadratic formula: x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} . But always try factoring first!

How do I check if I factored the quadratic correctly?

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Expand your factors back out! (x−3)(x+4)=x2+4x−3x−12=x2+x−12 (x-3)(x+4) = x^2 + 4x - 3x - 12 = x^2 + x - 12 ✓. If you get the original quadratic, your factoring is correct.

Can cubic equations have more than 3 solutions?

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No! A cubic equation can have at most 3 real solutions. In this problem, we found exactly 3: x = 0, 3, -4. Some cubics might have fewer real solutions due to complex roots.

Do I need to put my answers in any special order?

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While the mathematical solution is the same regardless of order, it's good practice to list them from smallest to largest: x = -4, 0, 3. This makes your answer easier to read and check.

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