Solve the Equation: 1/x + x² = 9/x in Rational Form

Q: How do I know when to take a cube root?

When you have $ x^3 = 8 $, you need to find what number cubed equals 8. Since 2 × 2 × 2 = 8, we know $ x = \sqrt[3]{8} = 2 $.

Q: Could there be other solutions besides x = 2?

For real numbers, x = 2 is the only solution. Cubic equations can have up to 3 solutions, but in this case, the other two solutions involve complex numbers, which are beyond basic algebra.

Q: What if I subtract 1/x from both sides instead?

That would give you $ x^2 = \frac{8}{x} $, which is harder to solve! Multiplying by x first eliminates all fractions at once, making the problem much simpler.

Q: How can I check my answer without a calculator?

Substitute x = 2: $ \frac{1}{2} + 2^2 = \frac{1}{2} + 4 = \frac{9}{2} $ and $ \frac{9}{2} = \frac{9}{2} $. The equation balances perfectly!

Cubic Equations with Fractional Terms

$\frac{1}{x}+x^2=\frac{9}{x}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Multiply by denominators to eliminate fractions

00:10 Simplify what's possible

00:17 Isolate X

00:26 Extract cube root

00:33 Break down 8 into 2 to the power of 3

00:38 Cube root cancels out power of 3

00:41 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

$\frac{1}{x}+x^2=\frac{9}{x}$

Step-by-step solution

To solve the equation $\frac{1}{x} + x^2 = \frac{9}{x}$ , we follow these steps:

Step 1: Eliminate the fractions by multiplying both sides of the equation by $x$ , yielding $1 + x^3 = 9$ .
Step 2: Rearrange the equation to standard form: $x^3 + 1 = 9$ . Simplify this to $x^3 = 8$ .
Step 3: Solve for $x$ by taking the cube root of both sides: $x = \sqrt[3]{8}$ or simply $x = 2$ .

Thus, the solution to the equation $\frac{1}{x} + x^2 = \frac{9}{x}$ is $x = 2$ .

Final Answer

$x=2$

Key Points to Remember

Essential concepts to master this topic

Elimination: Multiply both sides by x to clear all fractions
Technique: Rearrange $1 + x^3 = 9$ to get $x^3 = 8$
Check: Substitute x = 2: $\frac{1}{2} + 4 = \frac{9}{2}$ gives $\frac{9}{2} = \frac{9}{2}$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all terms by x
Don't multiply only the fractions by x and leave x² unchanged = wrong equation! This creates $1 + x^2 \cdot x = 9$ instead of the correct $1 + x^3 = 9$ . Always multiply every single term on both sides by x.

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 I multiply both sides by x when x could be zero?

Great observation! We can multiply by x because we're looking for solutions where x ≠ 0. Since the original equation has fractions with x in the denominator, x = 0 would make the equation undefined anyway.

How do I know when to take a cube root?

When you have $x^3 = 8$ , you need to find what number cubed equals 8. Since 2 × 2 × 2 = 8, we know $x = \sqrt[3]{8} = 2$ .

Could there be other solutions besides x = 2?

For real numbers, x = 2 is the only solution. Cubic equations can have up to 3 solutions, but in this case, the other two solutions involve complex numbers, which are beyond basic algebra.

What if I subtract 1/x from both sides instead?

That would give you $x^2 = \frac{8}{x}$ , which is harder to solve! Multiplying by x first eliminates all fractions at once, making the problem much simpler.

How can I check my answer without a calculator?

Substitute x = 2: $\frac{1}{2} + 2^2 = \frac{1}{2} + 4 = \frac{9}{2}$ and $\frac{9}{2} = \frac{9}{2}$ . The equation balances perfectly!

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Solve the Equation: 1/x + x² = 9/x in Rational Form

Cubic Equations with Fractional Terms

❤️ 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 I multiply both sides by x when x could be zero?

How do I know when to take a cube root?

Could there be other solutions besides x = 2?

What if I subtract 1/x from both sides instead?

How can I check my answer without a calculator?

🌟 Unlock Your Math Potential

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