Solve the Equation: x⁴ + 2x² = 0 Using Factor Method

Q: Why can't I solve x² = -2 for real numbers?

Because squares of real numbers are always non-negative! When you square any real number (positive, negative, or zero), the result is always ≥ 0. Since -2

Q: What if I tried to solve this without factoring?

You could substitute $ y = x^2 $ to get $ y^2 + 2y = 0 $, then factor as $ y(y + 2) = 0 $. This gives the same result but factoring directly is more efficient!

Q: Are there other ways to factor this equation?

Yes! You could write it as $ x^2(x^2 + 2) = 0 $ or factor step by step. The key is recognizing the common factor of $ x^2 $ in both terms.

Q: Why is x = 0 the only real solution?

From $ x^2(x^2 + 2) = 0 $, either $ x^2 = 0 $ (giving x = 0) or $ x^2 + 2 = 0 $ (giving $ x^2 = -2 $, which has no real solutions). So x = 0 is the only real answer!

Q: How do I check if x = 0 is correct?

Substitute back into the original equation: $ 0^4 + 2(0^2) = 0 + 2(0) = 0 + 0 = 0 $. Since we get 0 = 0, our answer is correct!

Quartic Equations with Zero Product Property

$x^4+2x^2=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Factor with the X² term

00:15 Take out the common factor from the parentheses

00:18 This is one solution that zeros the equation

00:26 Now let's check which solutions zero the second factor

00:30 Isolate X

00:34 Any number squared is necessarily positive, therefore there is no solution

00:38 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

$x^4+2x^2=0$

Step-by-step solution

To solve the equation $x^4 + 2x^2 = 0$ , we will use the technique of factoring. Let's proceed step-by-step:

First, notice that both terms $x^4$ and $2x^2$ have a common factor of $x^2$ . We can factor $x^2$ out from the equation:

$x^2(x^2 + 2) = 0$

Now, to solve for $x$ , we apply the Zero Product Property, which gives us that at least one of the factors must be zero:

$x^2 = 0$ or
$x^2 + 2 = 0$

Solving the first case, $x^2 = 0$ :

$x = 0$

For the second case, $x^2 + 2 = 0$ , we reach:

$x^2 = -2$

Since $x^2 = -2$ has no real solutions (squares of real numbers are non-negative), we can conclude that this equation doesn't provide additional real solutions.

Therefore, the only real solution to the given equation is $x = 0$ .

The correct choice from the provided options is:

$x=0$

Final Answer

$x=0$

Key Points to Remember

Essential concepts to master this topic

Factoring: Extract common factors before applying zero product property
Technique: Factor out $x^2$ from $x^4 + 2x^2 = x^2(x^2 + 2) = 0$
Check: Substitute x = 0: $0^4 + 2(0^2) = 0 + 0 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check for complex solutions
Don't ignore the factor $x^2 + 2 = 0$ completely = missing part of the analysis! While it gives no real solutions, recognizing this shows complete understanding. Always examine all factors even if they don't yield real solutions.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-8=x^2+4 $$

FAQ

Everything you need to know about this question

Why can't I solve x² = -2 for real numbers?

Because squares of real numbers are always non-negative! When you square any real number (positive, negative, or zero), the result is always ≥ 0. Since -2 < 0, there's no real number that squares to -2.

What if I tried to solve this without factoring?

You could substitute $y = x^2$ to get $y^2 + 2y = 0$ , then factor as $y(y + 2) = 0$ . This gives the same result but factoring directly is more efficient!

Are there other ways to factor this equation?

Yes! You could write it as $x^2(x^2 + 2) = 0$ or factor step by step. The key is recognizing the common factor of $x^2$ in both terms.

Why is x = 0 the only real solution?

From $x^2(x^2 + 2) = 0$ , either $x^2 = 0$ (giving x = 0) or $x^2 + 2 = 0$ (giving $x^2 = -2$ , which has no real solutions). So x = 0 is the only real answer!

How do I check if x = 0 is correct?

Substitute back into the original equation: $0^4 + 2(0^2) = 0 + 2(0) = 0 + 0 = 0$ . Since we get 0 = 0, our answer is correct!

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Solve the Equation: x⁴ + 2x² = 0 Using Factor Method

Quartic Equations 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 solve x² = -2 for real numbers?

What if I tried to solve this without factoring?

Are there other ways to factor this equation?

Why is x = 0 the only real solution?

How do I check if x = 0 is correct?

🌟 Unlock Your Math Potential

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