Solve (x+1)² = x² : Perfect Square Equation Challenge

Perfect Square Equations with Linear Solutions

Solve the following equation:

(x+1)2=x2 (x+1)^2=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 Let's use shortened multiplication formulas to open the parentheses
00:10 Calculate the multiplication and the square
00:16 Simplify what we can
00:23 Isolate X
00:32 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following equation:

(x+1)2=x2 (x+1)^2=x^2

2

Step-by-step solution

Let's examine the given equation:

(x+1)2=x2 (x+1)^2=x^2 First, let's simplify the equation, using the perfect square binomial formula:

(a±b)2=a2±2ab+b2 (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+1)2=x2↓x2+2⋅x⋅1+12=x2x2+2x+1=x22x=−1/:2x=−12 (x+1)^2=x^2 \\ \downarrow\\ x^2+2\cdot x\cdot1+1^2=x^2\\ x^2+2x+1= x^2\\ 2x=-1\hspace{6pt}\text{/}:2\\ \boxed{x=-\frac{1}{2}} Therefore, the correct answer is answer A.

3

Final Answer

x=−12 x=-\frac{1}{2}

Key Points to Remember

Essential concepts to master this topic
  • Formula: Apply (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 to expand
  • Technique: Expand (x+1)2=x2+2x+1 (x+1)^2 = x^2 + 2x + 1 then simplify
  • Check: Substitute x=−12 x = -\frac{1}{2} : both sides equal 14 \frac{1}{4} ✓

Common Mistakes

Avoid these frequent errors
  • Trying to take square root of both sides immediately
    Don't take √ of both sides right away = x+1=±x x+1 = ±x which gives complex cases! This skips the algebraic expansion and creates unnecessary complications. Always expand the perfect square first, then simplify the resulting linear equation.

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

Why can't I just take the square root of both sides?

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While that's a valid algebraic operation, it creates two cases to consider: x+1=x x+1 = x and x+1=−x x+1 = -x . Expanding first is simpler and leads directly to the answer!

How do I expand (x+1)2 (x+1)^2 ?

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Use the perfect square formula: (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 . So (x+1)2=x2+2(x)(1)+12=x2+2x+1 (x+1)^2 = x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1 .

What happens to the x2 x^2 terms?

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They cancel out! When you have x2+2x+1=x2 x^2 + 2x + 1 = x^2 , subtract x2 x^2 from both sides to get 2x+1=0 2x + 1 = 0 .

Is x=−12 x = -\frac{1}{2} really the only solution?

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Yes! After expanding and simplifying, we get a linear equation 2x=−1 2x = -1 , which has exactly one solution. Linear equations always have one unique solution.

How can I check my answer is correct?

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Substitute x=−12 x = -\frac{1}{2} into both sides: Left: (−12+1)2=(12)2=14 (-\frac{1}{2}+1)^2 = (\frac{1}{2})^2 = \frac{1}{4} . Right: (−12)2=14 (-\frac{1}{2})^2 = \frac{1}{4} . Both equal 14 \frac{1}{4} ✓

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