Solve the Reciprocal Equation: Finding X in 9/x = x

Q: Why can't x equal zero in this equation?

Because $ \frac{9}{x} $ means 9 divided by x. Division by zero is undefined in mathematics, so x = 0 is not allowed in the domain of this equation.

Q: How do I know when to multiply both sides by x?

When you see a fraction equal to a whole expression, multiplying both sides by the denominator eliminates the fraction. Just remember to check that the denominator isn't zero!

Q: Why does this become a quadratic equation?

After multiplying both sides by x, we get $ 9 = x^2 $. Any equation with $ x^2 $ as the highest power is a quadratic equation, even if it doesn't look like the standard $ ax^2 + bx + c = 0 $ form.

Q: Can I solve this by taking the square root directly?

Yes! From $ x^2 = 9 $, you can take the square root of both sides: $ x = \pm\sqrt{9} = \pm3 $. This is often faster than factoring!

Q: What if the number on the right wasn't a perfect square?

If you had $ \frac{8}{x} = x $, you'd get $ x^2 = 8 $, so $ x = \pm\sqrt{8} = \pm2\sqrt{2} $. Not all reciprocal equations have nice integer solutions!

Q: How do I verify both solutions work?

Substitute each solution back into the original equation:For x = 3: $ \frac{9}{3} = 3 $ ✓For x = -3: $ \frac{9}{-3} = -3 $ ✓Both sides equal, so both solutions are correct!

Reciprocal Equations with Quadratic Transformation

$\frac{9}{x}=x$

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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 the denominator to eliminate the fraction

00:08 Any number multiplied by itself is actually squared

00:11 Extract the root

00:14 When extracting a root there are always 2 solutions (positive, negative)

00:17 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

$\frac{9}{x}=x$

Step-by-step solution

To solve the equation $\frac{9}{x} = x$ , we will eliminate the fraction by multiplying both sides by $x$ , provided $x \neq 0$ .

Our steps are as follows:

Step 1: Multiply both sides by $x$ to obtain $9 = x^2$ .
Step 2: Rearrange the equation to standard quadratic form: $x^2 - 9 = 0$ .
Step 3: Recognize that this can be solved by factoring as a difference of squares: $(x - 3)(x + 3) = 0$ .
Step 4: Solve for $x$ by setting each factor to zero, giving $x - 3 = 0$ or $x + 3 = 0$ .
Step 5: Solve these equations to find $x = 3$ or $x = -3$ .

Thus, the solutions to the equation $\frac{9}{x} = x$ are $x = 3$ and $x = -3$ .

Therefore, the correct answer, which matches choice 3, is $x = \pm 3$ .

Final Answer

$x=\operatorname{\pm}3$

Key Points to Remember

Essential concepts to master this topic

Domain Restriction: x cannot equal zero since division by zero is undefined
Technique: Multiply both sides by x: $\frac{9}{x} \times x = x \times x$ gives $9 = x^2$
Check: Substitute x = 3: $\frac{9}{3} = 3$ and x = -3: $\frac{9}{-3} = -3$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to consider both positive and negative square roots
Don't solve $x^2 = 9$ and only write x = 3! This misses half the solution. The equation $x^2 = 9$ means x could be 3 or -3 since both $3^2 = 9$ and $(-3)^2 = 9$ . Always remember $x = \pm\sqrt{9} = \pm3$ .

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't x equal zero in this equation?

Because $\frac{9}{x}$ means 9 divided by x. Division by zero is undefined in mathematics, so x = 0 is not allowed in the domain of this equation.

How do I know when to multiply both sides by x?

When you see a fraction equal to a whole expression, multiplying both sides by the denominator eliminates the fraction. Just remember to check that the denominator isn't zero!

Why does this become a quadratic equation?

After multiplying both sides by x, we get $9 = x^2$ . Any equation with $x^2$ as the highest power is a quadratic equation, even if it doesn't look like the standard $ax^2 + bx + c = 0$ form.

Can I solve this by taking the square root directly?

Yes! From $x^2 = 9$ , you can take the square root of both sides: $x = \pm\sqrt{9} = \pm3$ . This is often faster than factoring!

What if the number on the right wasn't a perfect square?

If you had $\frac{8}{x} = x$ , you'd get $x^2 = 8$ , so $x = \pm\sqrt{8} = \pm2\sqrt{2}$ . Not all reciprocal equations have nice integer solutions!

How do I verify both solutions work?

Substitute each solution back into the original equation:

For x = 3: $\frac{9}{3} = 3$ ✓
For x = -3: $\frac{9}{-3} = -3$ ✓

Both sides equal, so both solutions are correct!

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Solve the Reciprocal Equation: Finding X in 9/x = x

Reciprocal Equations with Quadratic Transformation

❤️ 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 x equal zero in this equation?

How do I know when to multiply both sides by x?

Why does this become a quadratic equation?

Can I solve this by taking the square root directly?

What if the number on the right wasn't a perfect square?

How do I verify both solutions work?

🌟 Unlock Your Math Potential

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