Solve (x+2)² - 12 = x²: Perfect Square Equation Challenge

Q: What's the binomial expansion formula I should use?

Use $ (a+b)^2 = a^2 + 2ab + b^2 $. For $ (x+2)^2 $: a = x and b = 2, so you get $ x^2 + 2(x)(2) + 2^2 = x^2 + 4x + 4 $.

Q: Why can I cancel the x² terms from both sides?

Because they're identical terms on both sides! When you have $ x^2 + 4x - 8 = x^2 $, subtracting $ x^2 $ from both sides eliminates them completely.

Q: How do I know when to use the perfect square formula?

Look for expressions like $ (x+a)^2 $ or $ (x-a)^2 $ where something is squared. These always need the binomial expansion formula before you can solve!

Q: What if I expand incorrectly and get the wrong answer?

Always check your work! Substitute your answer back into the original equation. If $ (x+2)^2 - 12 $ doesn't equal $ x^2 $, you made an expansion error.

Q: Can I solve this equation a different way?

You could rearrange to get $ (x+2)^2 - x^2 = 12 $, then use difference of squares, but expanding the perfect square is usually the most straightforward method.

Quadratic Expansion with Algebraic Simplification

Solve the following equation:

$(x+2)^2-12=x^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Use factoring formulas to open the parentheses

00:13 Calculate the products

00:24 Simplify what we can

00:30 Collect like terms

00:34 Isolate X

00:49 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$(x+2)^2-12=x^2$

Step-by-step solution

Let's examine the given equation:

$(x+2)^2-12=x^2$ First, let's simplify the equation, for this we'll make use of the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$ ,

We'll start by opening the parentheses on the left side using the perfect square formula and then move terms and combine like terms, in the final step we'll solve the simplified equation we obtain:

$(x+2)^2-12=x^2 \\ \downarrow\\ x^2+2\cdot x\cdot2+2^2-12=x^2\\ x^2+4x+4-12= x^2\\ 4x=8\hspace{6pt}\text{/}:4\\ \boxed{x=2}$ Therefore, the correct answer is answer C.

Final Answer

$x=2$

Key Points to Remember

Essential concepts to master this topic

Perfect Square Rule: $(x+2)^2 = x^2 + 4x + 4$ using binomial formula
Simplification: Cancel $x^2$ terms: $x^2 + 4x + 4 - 12 = x^2$ becomes $4x = 8$
Verification: Substitute x = 2: $(2+2)^2 - 12 = 16 - 12 = 4 = 2^2$ ✓

Common Mistakes

Avoid these frequent errors

Not expanding the perfect square correctly
Don't just square the first and last terms: $(x+2)^2 = x^2 + 4$ = wrong answer! This ignores the middle term and gives $x = -4$ . Always use the complete binomial formula: $(a+b)^2 = a^2 + 2ab + b^2$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-8=x^2+4 $$

FAQ

Everything you need to know about this question

What's the binomial expansion formula I should use?

Use $(a+b)^2 = a^2 + 2ab + b^2$ . For $(x+2)^2$ : a = x and b = 2, so you get $x^2 + 2(x)(2) + 2^2 = x^2 + 4x + 4$ .

Why can I cancel the x² terms from both sides?

Because they're identical terms on both sides! When you have $x^2 + 4x - 8 = x^2$ , subtracting $x^2$ from both sides eliminates them completely.

How do I know when to use the perfect square formula?

Look for expressions like $(x+a)^2$ or $(x-a)^2$ where something is squared. These always need the binomial expansion formula before you can solve!

What if I expand incorrectly and get the wrong answer?

Always check your work! Substitute your answer back into the original equation. If $(x+2)^2 - 12$ doesn't equal $x^2$ , you made an expansion error.

Can I solve this equation a different way?

You could rearrange to get $(x+2)^2 - x^2 = 12$ , then use difference of squares, but expanding the perfect square is usually the most straightforward method.

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Solve (x+2)² - 12 = x²: Perfect Square Equation Challenge

Quadratic Expansion with Algebraic Simplification

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What's the binomial expansion formula I should use?

Why can I cancel the x² terms from both sides?

How do I know when to use the perfect square formula?

What if I expand incorrectly and get the wrong answer?

Can I solve this equation a different way?

🌟 Unlock Your Math Potential

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