Solve the Fraction Equation: x^2/3 + 4/3 = 3(x - 2)(x + 2)

Q: Why do I multiply both sides by 3 instead of dividing by the fraction?

Multiplying by 3 is much easier and cleaner than dividing by a fraction! It eliminates the denominator completely, giving you $ x^2 + 4 = 9(x-2)(x+2) $ to work with.

Q: How do I recognize when to use the difference of squares formula?

Look for the pattern (a-b)(a+b)! In this problem, $ (x-2)(x+2) $ has the same variable with opposite signs, so it equals $ x^2 - 4 $.

Q: Why do I get two answers for x?

When you solve $ x^2 = 5 $, you need both positive and negative square roots! Both $ +\sqrt{5} $ and $ -\sqrt{5} $ make the equation true when squared.

Q: How can I check if both solutions are correct?

Substitute each answer back into the original equation separately. Both $ x = \sqrt{5} $ and $ x = -\sqrt{5} $ should make both sides equal the same value!

Q: What if I can't simplify the square root?

That's perfectly normal! $ \sqrt{5} $ cannot be simplified further because 5 has no perfect square factors other than 1. Leave it as $ \pm\sqrt{5} $.

Quadratic Equations with Fraction Coefficients

Resolve:

$\frac{x^2+4}{3}=3(x-2)(x+2)$

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Understand the problem

Resolve:

$\frac{x^2+4}{3}=3(x-2)(x+2)$

Step-by-step solution

To solve the equation $\frac{x^2 + 4}{3} = 3(x-2)(x+2)$ , we will follow these steps:

Step 1: Remove the fraction by multiplying both sides by 3.
Step 2: Expand and simplify the right side using the difference of squares.
Step 3: Rearrange into a standard quadratic form.
Step 4: Solve the quadratic equation.

Let's begin:

Step 1: Multiply both sides by 3 to eliminate the fraction:

x^2 + 4 = 9(x-2)(x+2)

Step 2: Recognize that $(x-2)(x+2)$ is a difference of squares:

(x-2)(x+2) = x^2 - 4

Then we have:

x^2 + 4 = 9(x^2 - 4)

Step 3: Distribute the 9 on the right side:

x^2 + 4 = 9x^2 - 36

Step 4: Rearrange this into a standard quadratic form:

x^2 - 9x^2 = -36 - 4

Simplify:

-8x^2 = -40

Divide everything by -8 to solve for $x^2$ :

x^2 = 5

Step 5: Solve by taking the square root of both sides:

x = \pm \sqrt{5}

Therefore, the solution to the problem is $x = \pm \sqrt{5}$ .

Final Answer

$\operatorname{\pm}\sqrt{5}$

Key Points to Remember

Essential concepts to master this topic

Rule: Multiply both sides by LCD to eliminate fraction denominators
Technique: Use difference of squares: $(x-2)(x+2) = x^2 - 4$
Check: Substitute $x = \sqrt{5}$ back: $\frac{5+4}{3} = 3(5-4)$ gives 3 = 3 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute multiplication across all terms
Don't multiply just the fraction by 3 and leave (x-2)(x+2) unchanged = wrong equation setup! This creates an unbalanced equation with mixed operations. Always multiply every single term on both sides by the same number.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why do I multiply both sides by 3 instead of dividing by the fraction?

Multiplying by 3 is much easier and cleaner than dividing by a fraction! It eliminates the denominator completely, giving you $x^2 + 4 = 9(x-2)(x+2)$ to work with.

How do I recognize when to use the difference of squares formula?

Look for the pattern (a-b)(a+b)! In this problem, $(x-2)(x+2)$ has the same variable with opposite signs, so it equals $x^2 - 4$ .

Why do I get two answers for x?

When you solve $x^2 = 5$ , you need both positive and negative square roots! Both $+\sqrt{5}$ and $-\sqrt{5}$ make the equation true when squared.

How can I check if both solutions are correct?

Substitute each answer back into the original equation separately. Both $x = \sqrt{5}$ and $x = -\sqrt{5}$ should make both sides equal the same value!

What if I can't simplify the square root?

That's perfectly normal! $\sqrt{5}$ cannot be simplified further because 5 has no perfect square factors other than 1. Leave it as $\pm\sqrt{5}$ .

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