Solve the Equation: (x+2)² - 12 = x²

Quadratic Expansion with Canceling Terms

(x+2)2−12=x2 (x+2)^2-12=x^2

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:05 Let's find X. Ready?
00:08 First, open the parentheses using multiplication rules. Let's go step by step.
00:15 Now, solve the multiplications and calculate the squares. Keep going, you're doing great!
00:21 Next, simplify everything you can. Let's make it easier.
00:26 It's time to isolate X. Almost there!
00:40 And that's how we find the solution. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(x+2)2−12=x2 (x+2)^2-12=x^2

2

Step-by-step solution

Let's solve the equation. First, we'll simplify the algebraic expressions using the perfect square binomial formula:

(a±b)2=a2±2ab+b2 (a\pm b)^2=a^2\pm2ab+b^2 We'll apply the mentioned formula and expand the parentheses in the expressions in the equation:

(x+2)2−12=x2x2+2⋅x⋅2+22−12=x2x2+4x+4−12=x2 (x+2)^2-12=x^2 \\ x^2+2\cdot x\cdot2+2^2-12=x^2 \\ x^2+4x+4-12=x^2 \\ We'll continue and combine like terms, by moving terms between sides. Then we can notice 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:

x2+4x+4−12=x24x=8/:4x=2 x^2+4x+4-12=x^2 \\ 4x=8\hspace{8pt}\text{/}:4\\ \boxed{x=2} Therefore, the correct answer is answer B.

3

Final Answer

x=2 x=2

Key Points to Remember

Essential concepts to master this topic
  • Perfect Square Formula: (x+2)2=x2+4x+4 (x+2)^2 = x^2 + 4x + 4 using binomial expansion
  • Technique: Expand left side: x2+4x+4−12=x2 x^2 + 4x + 4 - 12 = x^2
  • Check: Substitute x = 2: (2+2)2−12=16−12=4=22 (2+2)^2 - 12 = 16 - 12 = 4 = 2^2 ✓

Common Mistakes

Avoid these frequent errors
  • Not expanding the perfect square correctly
    Don't write (x+2)2=x2+4 (x+2)^2 = x^2 + 4 = wrong answer! This skips the middle term and gives incorrect solutions. Always use the complete formula: (x+2)2=x2+4x+4 (x+2)^2 = x^2 + 4x + 4 .

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:


\( (x+3)^2 \)

FAQ

Everything you need to know about this question

Why does the x2 x^2 term disappear?

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Great observation! When you move x2 x^2 from the right side to the left, it cancels out with the x2 x^2 term from the expansion. This turns a seemingly quadratic equation into a simple linear one!

Do I always need to expand (x+2)2 (x+2)^2 ?

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Yes! You must expand using the perfect square formula: (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 . So (x+2)2=x2+4x+4 (x+2)^2 = x^2 + 4x + 4 . Never just square each term separately.

How do I remember the perfect square formula?

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Memory tip: Think "First squared, plus twice the product, plus last squared". For (x+2)2 (x+2)^2 : x2+2(x)(2)+22 x^2 + 2(x)(2) + 2^2

What if I get a different answer?

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Double-check your expansion! The most common error is forgetting the 4x 4x term. Make sure you have: x2+4x+4−12=x2 x^2 + 4x + 4 - 12 = x^2 , which simplifies to 4x−8=0 4x - 8 = 0 .

Can this equation have two solutions?

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No! Even though it looks quadratic initially, the x2 x^2 terms cancel out, leaving us with 4x=8 4x = 8 . Linear equations have exactly one solution.

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