Solve the Fraction Equation: Find x in (x²-9)/(x-3) = 0

Q: How do I remember which solutions to exclude?

Always check each potential solution in the original denominator. If substituting a value makes any denominator equal zero, that value must be excluded from your final answer, even if it solves the numerator equation.

Q: What's the difference between x²-9=0 and (x²-9)/(x-3)=0?

The equation $ x^2-9=0 $ has solutions x = 3 and x = -3. But $ \frac{x^2-9}{x-3}=0 $ excludes x = 3 because it makes the denominator zero. Domain restrictions change the solution set!

Q: Should I simplify the fraction before solving?

You can, but be careful! $ \frac{x^2-9}{x-3} = \frac{(x-3)(x+3)}{x-3} = x+3 $ when x ≠ 3. However, you must still remember that x = 3 is not allowed in the original equation.

Rational Equations with Domain Restrictions

Resolve:

$\frac{x^2-9}{x-3}=0$

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Understand the problem

Resolve:

$\frac{x^2-9}{x-3}=0$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Factor the numerator using the difference of squares formula.
Step 2: Set the factored numerator equal to zero to find potential solutions for $x$ .
Step 3: Ensure none of these solutions result in the denominator being zero.

Now, let's work through each step:
Step 1: The numerator $x^2 - 9$ can be factored as $(x - 3)(x + 3)$ .
Step 2: Set the factored numerator to zero: $(x - 3)(x + 3) = 0$ .
This gives two potential solutions: $x - 3 = 0$ or $x + 3 = 0$ . Solving these equations, we get $x = 3$ or $x = -3$ .
Step 3: Verify that these solutions do not result in division by zero:
- For $x = 3$ , the denominator $x - 3 = 0$ , which means division by zero occurs, so it is not a valid solution.
- For $x = -3$ , the denominator $x - 3$ is not zero, as $(-3 - 3) = -6$ , hence $x = -3$ is a valid solution.

Therefore, the solution to the problem is $x = -3$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: For rational equations, numerator equals zero while denominator stays nonzero
Technique: Factor $x^2-9 = (x-3)(x+3) = 0$ gives x = ±3
Check: Verify x = -3 doesn't make denominator zero: (-3-3) = -6 ≠ 0 ✓

Common Mistakes

Avoid these frequent errors

Including solutions that make the denominator zero
Don't accept x = 3 as a solution even though it makes the numerator zero = undefined expression! When x = 3, the denominator x - 3 = 0, creating division by zero. Always check that solutions don't make any denominator equal to zero.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why can't x = 3 be a solution if it makes the numerator zero?

While x = 3 does make the numerator zero, it also makes the denominator zero! This creates 0/0, which is undefined in mathematics. A fraction can only equal zero when the numerator is zero and the denominator is nonzero.

How do I remember which solutions to exclude?

Always check each potential solution in the original denominator. If substituting a value makes any denominator equal zero, that value must be excluded from your final answer, even if it solves the numerator equation.

What's the difference between x²-9=0 and (x²-9)/(x-3)=0?

The equation $x^2-9=0$ has solutions x = 3 and x = -3. But $\frac{x^2-9}{x-3}=0$ excludes x = 3 because it makes the denominator zero. Domain restrictions change the solution set!

Can a rational equation have no solutions?

Yes! If all solutions to the numerator equation also make the denominator zero, then the rational equation has no solutions. Always check every potential solution against the domain restrictions.

Should I simplify the fraction before solving?

You can, but be careful! $\frac{x^2-9}{x-3} = \frac{(x-3)(x+3)}{x-3} = x+3$ when x ≠ 3. However, you must still remember that x = 3 is not allowed in the original equation.

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