Solve (x-3)(x+3) = (x-9)(x+9)+x+5: Factored Form Equation Challenge

Q: Why does the x² term disappear when solving?

Both sides have $ x^2 $ terms, so when you subtract $ x^2 $ from both sides, they cancel out completely! This turns a quadratic-looking equation into a simple linear equation.

Q: How do I know when to use difference of squares?

Look for the pattern (a-b)(a+b) where you have the same terms with opposite signs. Examples: (x-5)(x+5), (2x-7)(2x+7), or (x-9)(x+9).

Q: What if I made an arithmetic error in the final steps?

Double-check your arithmetic: $ -9 = x - 76 $ means $ x = -9 + 76 = 67 $. Common error is forgetting to add 76 to both sides properly!

Q: Should I expand everything first instead?

No! Expanding creates more work and opportunities for mistakes. The difference of squares formula $ (a-b)(a+b) = a^2 - b^2 $ is much faster and more reliable.

Q: How can I check if 67 is really correct?

Substitute back: Left side = $ (67-3)(67+3) = 64 \times 70 = 4480 $. Right side = $ (67-9)(67+9) + 67 + 5 = 58 \times 76 + 72 = 4408 + 72 = 4480 $ ✓

Difference of Squares with Linear Terms

Solve the following equation:

_{^{$(x-3)(x+3)=(x-9)(x+9)+x+5$}}

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:05 Let's use the shortened multiplication formulas

00:25 Calculate 9 squared

00:31 Simplify what we can

00:38 Group terms

00:44 Isolate X

00:52 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

Solve the following equation:

_{^{$(x-3)(x+3)=(x-9)(x+9)+x+5$}}

Step-by-step solution

To solve this problem, we'll use the following steps:

Step 1: Recognize and apply the difference of squares formula to both sides of the equation.
Step 2: Simplify the equation.
Step 3: Solve for $x$ .

Let's go through the solution step-by-step:

Step 1: Simplify each side using the difference of squares formula.

The left-hand side is $(x-3)(x+3) = x^2 - 3^2 = x^2 - 9$ .

The right-hand side is $(x-9)(x+9) + x + 5 = (x^2 - 9^2) + x + 5 = x^2 - 81 + x + 5$ .

Step 2: Set the simplified expressions equal to each other.

$x^2 - 9 = x^2 - 81 + x + 5$

Step 3: Subtract $x^2$ from both sides to eliminate the quadratic term, simplifying the equation:

$-9 = -81 + x + 5$

Simplify by combining like terms on the right-hand side:

$-9 = x - 76$

Add 76 to both sides to solve for $x$ :

$-9 + 76 = x$

$x = 67$

Therefore, the solution to the equation is $\mathbf{x = 67}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Apply (a-b)(a+b) = a² - b² formula first
Simplification: Left side: x² - 9, Right side: x² - 81 + x + 5
Verification: Substitute x = 67: (64)(70) = (-58)(76) + 72 ✓

Common Mistakes

Avoid these frequent errors

Expanding products instead of using difference of squares
Don't expand (x-3)(x+3) as x² - 3x + 3x - 9 = wrong approach! This wastes time and increases error chances. Always recognize and apply the difference of squares formula (a-b)(a+b) = a² - b² directly.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why does the x² term disappear when solving?

Both sides have $x^2$ terms, so when you subtract $x^2$ from both sides, they cancel out completely! This turns a quadratic-looking equation into a simple linear equation.

How do I know when to use difference of squares?

Look for the pattern (a-b)(a+b) where you have the same terms with opposite signs. Examples: (x-5)(x+5), (2x-7)(2x+7), or (x-9)(x+9).

What if I made an arithmetic error in the final steps?

Double-check your arithmetic: $-9 = x - 76$ means $x = -9 + 76 = 67$ . Common error is forgetting to add 76 to both sides properly!

Should I expand everything first instead?

No! Expanding creates more work and opportunities for mistakes. The difference of squares formula $(a-b)(a+b) = a^2 - b^2$ is much faster and more reliable.

How can I check if 67 is really correct?

Substitute back: Left side = $(67-3)(67+3) = 64 \times 70 = 4480$ . Right side = $(67-9)(67+9) + 67 + 5 = 58 \times 76 + 72 = 4408 + 72 = 4480$ ✓

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Short Multiplication Formulas questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime