Solve Complex Fraction Equation: (x²-64)/(x-8) = 17(x+8)/(64-x²)

Q: Why can't x equal 8 or -8 in this problem?

These values make the denominators zero, which creates undefined expressions. When $ x = 8 $, we get $ \frac{something}{0} $ which is undefined in mathematics.

Q: How do I know when to factor x² - 64?

Recognize the difference of squares pattern: $ a^2 - b^2 = (a-b)(a+b) $. Since $ x^2 - 64 = x^2 - 8^2 $, it factors as $ (x-8)(x+8) $.

Q: What's the difference between 64 - x² and x² - 64?

They're opposites! Notice that $ 64 - x^2 = -(x^2 - 64) = -(x-8)(x+8) $. This negative sign is crucial for simplifying the equation correctly.

Q: Why does the answer involve square roots?

After simplifying and cross-multiplying, we get $ x^2 = 47 $. Since 47 isn't a perfect square, we need $ x = \pm\sqrt{47} $ to solve it.

Q: How do I verify this answer works?

Substitute $ x = \sqrt{47} $ back into the original equation. Both sides should give the same decimal value when calculated. Since $ \sqrt{47} \neq 8 $ and $ \sqrt{47} \neq -8 $, the solution is valid!

Rational Equations with Factored Denominators

$\frac{x^2-64}{x-8}=\frac{17(x+8)}{64-x^2}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's solve this problem together.

00:14 First, change 64 to 8 times 8, or 8 squared.

00:31 Now, let's use some helpful multiplication shortcuts.

01:07 Next step, simplify what we can to make it easier.

01:25 Multiply to get rid of the fraction, just like this.

01:45 Carefully open the parentheses and handle each term.

01:59 Rearrange the equation so the right side is zero.

02:20 Again, use the multiplication shortcuts to simplify.

02:27 Now, find both possible solutions to the equation.

02:54 And that's how we solve this question. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{x^2-64}{x-8}=\frac{17(x+8)}{64-x^2}$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Factorize the terms involving squares. Notice that $x^2 - 64 = (x-8)(x+8)$ and $64 - x^2 = (8-x)(8+x)$ .
Step 2: Rewrite both sides of the equation using these factorizations:

$\frac{(x-8)(x+8)}{x-8} = \frac{17(x+8)}{(8-x)(8+x)}$

Upon simplifying, the left side becomes $x+8$ because the $(x-8)$ term cancels out, as long as $x \neq 8$ .

$x + 8 = \frac{17(x+8)}{-(x-8)(x+8)}$

Step 3: Cancel $(x+8)$ from both numerator and denominator on the right side (assuming $x \neq -8$ ).
Step 4: This simplifies to:

$x + 8 = \frac{-17}{x-8}$

Step 5: Multiply both sides by $(x-8)$ to eliminate the denominator:

$(x+8)(x-8) = -17$

$x^2 - 64 = -17$

Step 6: Rearrange and solve for $x$ :

$x^2 = 47$

$x = \pm \sqrt{47}$

Therefore, the solution to the problem is $x = \pm \sqrt{47}$ .

Final Answer

$±\sqrt{47}$

Key Points to Remember

Essential concepts to master this topic

Rule: Factor difference of squares using $a^2 - b^2 = (a-b)(a+b)$
Technique: Cancel common factors: $\frac{(x-8)(x+8)}{x-8} = x+8$ when $x \neq 8$
Check: Verify $x = \pm\sqrt{47}$ doesn't make denominators zero ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check domain restrictions
Don't cancel factors without noting restrictions like x ≠ 8 and x ≠ -8 = invalid solutions! These values make denominators zero, creating undefined expressions. Always identify and exclude values that make any denominator zero before solving.

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 equal 8 or -8 in this problem?

These values make the denominators zero, which creates undefined expressions. When $x = 8$ , we get $\frac{something}{0}$ which is undefined in mathematics.

How do I know when to factor x² - 64?

Recognize the difference of squares pattern: $a^2 - b^2 = (a-b)(a+b)$ . Since $x^2 - 64 = x^2 - 8^2$ , it factors as $(x-8)(x+8)$ .

What's the difference between 64 - x² and x² - 64?

They're opposites! Notice that $64 - x^2 = -(x^2 - 64) = -(x-8)(x+8)$ . This negative sign is crucial for simplifying the equation correctly.

Why does the answer involve square roots?

After simplifying and cross-multiplying, we get $x^2 = 47$ . Since 47 isn't a perfect square, we need $x = \pm\sqrt{47}$ to solve it.

How do I verify this answer works?

Substitute $x = \sqrt{47}$ back into the original equation. Both sides should give the same decimal value when calculated. Since $\sqrt{47} \neq 8$ and $\sqrt{47} \neq -8$ , the solution is valid!

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