Solve the Equation x⁴ - 8x = 0: Finding All Solutions

Q: Why do I need to factor out x first?

Factoring out the greatest common factor (x) simplifies the equation and reveals one solution immediately! Without factoring, you'd have to work with the difficult quartic equation directly.

Q: How do I know when I have all the solutions?

Count the degree of your polynomial. A quartic equation (degree 4) can have up to 4 solutions, but some may be repeated or complex. In this case, we found 2 real solutions.

Q: What if I can't factor x³ - 8 easily?

Remember that 8 = 2³, so you're looking for the cube root of 8. You can also recognize this as a difference of cubes pattern if you want to factor further.

Q: Can I solve this by moving terms to one side?

The equation is already set equal to zero, which is perfect for factoring! Moving terms around would make it harder. Always look for factoring opportunities when one side equals zero.

Q: Why is x = 0 always a solution when I factor out x?

When you factor out x, you get $ x \cdot (\text{other factor}) = 0 $. By the zero product property, if x = 0, then the entire left side equals zero regardless of the other factor.

Quartic Equations with Factoring Methods

$x^4-8x=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Factor with term X

00:10 Take out the common factor from parentheses

00:23 This is one solution that zeroes the equation

00:27 Now let's solve the second term

00:31 Isolate X

00:36 Extract the cube root

00:49 Break down 8 into 2 cubed

00:52 Simplify what we can

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

$x^4-8x=0$

Step-by-step solution

To solve the equation $x^4 - 8x = 0$ , we'll follow these steps:

Step 1: Factor out the greatest common factor.
Step 2: Set each factor equal to zero and solve for $x$ .

Now, let's work through each step:
Step 1: The equation given is $x^4 - 8x = 0$ . Both terms on the left contain $x$ as a factor. We can factor out $x$ to rewrite the equation as:

$x(x^3 - 8) = 0$

Step 2: To find the solutions, set each factor to zero.

If $x = 0$ , then one solution is:

$x = 0$

Next, solve for $x$ in the equation $x^3 - 8 = 0$ :
Add 8 to both sides:

$x^3 = 8$

Take the cube root of both sides:

$x = \sqrt[3]{8} = 2$

Therefore, the solutions to the equation $x^4 - 8x = 0$ are $x = 0$ and $x = 2$ .

Thus, the correct answer is: $x = 0, 2$ .

Final Answer

$x=0,2$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find greatest common factor first before other methods
Technique: Factor out x to get x(x³ - 8) = 0
Check: Substitute both solutions: 0⁴ - 8(0) = 0 and 2⁴ - 8(2) = 0 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to factor completely
Don't stop at x(x³ - 8) = 0 without solving x³ - 8 = 0 = missing the solution x = 2! This gives you only half the answers. Always solve each factor completely and find all possible solutions.

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 need to factor out x first?

Factoring out the greatest common factor (x) simplifies the equation and reveals one solution immediately! Without factoring, you'd have to work with the difficult quartic equation directly.

How do I know when I have all the solutions?

Count the degree of your polynomial. A quartic equation (degree 4) can have up to 4 solutions, but some may be repeated or complex. In this case, we found 2 real solutions.

What if I can't factor x³ - 8 easily?

Remember that 8 = 2³, so you're looking for the cube root of 8. You can also recognize this as a difference of cubes pattern if you want to factor further.

Can I solve this by moving terms to one side?

The equation is already set equal to zero, which is perfect for factoring! Moving terms around would make it harder. Always look for factoring opportunities when one side equals zero.

Why is x = 0 always a solution when I factor out x?

When you factor out x, you get $x \cdot (\text{other factor}) = 0$ . By the zero product property, if x = 0, then the entire left side equals zero regardless of the other factor.

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Solve the Equation x⁴ - 8x = 0: Finding All Solutions

Quartic Equations with Factoring Methods

❤️ 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 do I need to factor out x first?

How do I know when I have all the solutions?

What if I can't factor x³ - 8 easily?

Can I solve this by moving terms to one side?

Why is x = 0 always a solution when I factor out x?

🌟 Unlock Your Math Potential

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