Solve the Equation: (x+3)(x-3) = x²+x Step by Step

Q: Why does the x² term cancel out completely?

After expanding both sides, you get $ x^2 - 9 = x^2 + x $. When you subtract $ x^2 $ from both sides, the quadratic terms disappear, leaving a simple linear equation!

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

Look for the pattern $ (a+b)(a-b) $ where you're multiplying the sum and difference of the same two terms. Here, we have $ (x+3)(x-3) $ - same variable and number, just different signs!

Q: What if I forgot the difference of squares formula?

You can still solve by multiplying term by term: $ x \cdot x + x \cdot (-3) + 3 \cdot x + 3 \cdot (-3) $. The middle terms $ -3x + 3x $ will cancel, giving you the same result!

Q: Why is x = -9 the only solution?

Since the $ x^2 $ terms canceled out, we're left with a linear equation $ -9 = x $. Linear equations have exactly one solution, unlike quadratic equations which can have two!

Algebraic Equations with Difference of Squares

$(x+3)(x-3)=x^2+x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's solve this problem together!

00:13 We'll use some multiplication shortcuts to expand the parentheses. Ready?

00:21 First, calculate three to the power of two. That's nine.

00:26 Now, let's group like terms and simplify as much as we can.

00:30 And there you have it! We've solved the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(x+3)(x-3)=x^2+x$

Step-by-step solution

Let's solve the equation. First, we'll simplify the algebraic expressions using the difference of squares formula:

$(a+b)(a-b)=a^2-b^2$ We'll apply this formula and expand the parentheses in the expressions in the equation:

$(x+3)(x-3)=x^2+x \\ x^2-3^2=x^2+x \\ x^2-9=x^2+x$ We'll continue and combine like terms. After moving terms around, we can see that the squared term cancels out, therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^2-9=x^2+x \\ -9=x\\ \downarrow\\ \boxed{x=-9}$ Therefore, the correct answer is answer B.

Final Answer

$x=-9$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply difference of squares formula: $(a+b)(a-b) = a^2 - b^2$
Technique: Expand $(x+3)(x-3) = x^2 - 9$ then solve linear equation
Check: Substitute $x = -9$ : $(-6)(-12) = 72$ and $81 - 9 = 72$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly expanding the left side term by term
Don't multiply (x+3)(x-3) as x² + 3x - 3x - 9 = x² + 6x - 9! This ignores the special difference of squares pattern and creates extra terms. Always recognize (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 cancel out completely?

After expanding both sides, you get $x^2 - 9 = x^2 + x$ . When you subtract $x^2$ from both sides, the quadratic terms disappear, leaving a simple linear equation!

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

Look for the pattern $(a+b)(a-b)$ where you're multiplying the sum and difference of the same two terms. Here, we have $(x+3)(x-3)$ - same variable and number, just different signs!

What if I forgot the difference of squares formula?

You can still solve by multiplying term by term: $x \cdot x + x \cdot (-3) + 3 \cdot x + 3 \cdot (-3)$ . The middle terms $-3x + 3x$ will cancel, giving you the same result!

Why is x = -9 the only solution?

Since the $x^2$ terms canceled out, we're left with a linear equation $-9 = x$ . Linear equations have exactly one solution, unlike quadratic equations which can have two!

Can I check my answer without substituting?

Substitution is the most reliable way! But you can also check your algebra: if $x^2 - 9 = x^2 + x$ simplifies to $-9 = x$ , then x must equal -9.

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