Solve the Equation: (x-1)² = x²

Quadratic Equations with Canceling Terms

(x−1)2=x2 (x-1)^2=x^2

❤️ 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
00:06 Let's start by finding the value of X.
00:10 We'll use the shorter multiplication formulas to open the brackets. Take each step slowly.
00:19 Now, let's simplify everything we can. Keep going, you're doing great!
00:25 Next, we need to isolate the variable X. You're almost there!
00:35 And that's how we solve this problem. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(x−1)2=x2 (x-1)^2=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 this formula and expand the parentheses in the expressions in the equation:

(x−1)2=x2x2−2⋅x⋅1+12=x2x2−2x+1=x2 (x-1)^2=x^2 \\ x^2-2\cdot x\cdot1+1^2=x^2 \\ x^2-2x+1=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:

x2−2x+1=x2−2x=−1/:(−2)x=12 x^2-2x+1=x^2 \\ -2x=-1\hspace{8pt}\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
  • Perfect Square Formula: (a−b)2=a2−2ab+b2 (a-b)^2 = a^2 - 2ab + b^2 for expansion
  • Technique: Expand (x−1)2=x2−2x+1 (x-1)^2 = x^2 - 2x + 1 then subtract x2 x^2 from both sides
  • Check: Substitute x=12 x = \frac{1}{2} : (12−1)2=(−12)2=14=(12)2 (\frac{1}{2}-1)^2 = (-\frac{1}{2})^2 = \frac{1}{4} = (\frac{1}{2})^2 ✓

Common Mistakes

Avoid these frequent errors
  • Taking square roots of both sides without expanding first
    Don't take (x−1)2=x2 \sqrt{(x-1)^2} = \sqrt{x^2} to get x−1=x x-1 = x = no solution! This ignores the ±sign rule and misses the algebraic structure. Always expand the perfect square first using (a−b)2=a2−2ab+b2 (a-b)^2 = a^2 - 2ab + b^2 .

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

\( (7b-3x)^2 \)

FAQ

Everything you need to know about this question

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

+

Taking square roots gives you ∣x−1∣=∣x∣ |x-1| = |x| , which creates a more complex absolute value equation. Expanding first using the perfect square formula is much simpler and avoids case analysis.

How did the x2 x^2 terms disappear?

+

When you expand (x−1)2=x2−2x+1 (x-1)^2 = x^2 - 2x + 1 , you get x2−2x+1=x2 x^2 - 2x + 1 = x^2 . Subtracting x2 x^2 from both sides cancels them out, leaving −2x+1=0 -2x + 1 = 0 .

Is this really a quadratic equation if it becomes linear?

+

Yes! It starts as a quadratic because of the squared terms, but the x2 x^2 terms cancel out. This is called a degenerate quadratic - it looks quadratic but simplifies to linear.

What if I expanded wrong and got a different answer?

+

Double-check your expansion: (x−1)2=x2−2x+1 (x-1)^2 = x^2 - 2x + 1 . The middle term is -2x (not -x or +2x). Use the pattern (a−b)2=a2−2ab+b2 (a-b)^2 = a^2 - 2ab + b^2 carefully.

Can I solve this by moving everything to one side first?

+

You could rearrange to (x−1)2−x2=0 (x-1)^2 - x^2 = 0 , but you'll still need to expand the perfect square. The approach shown (expand first, then simplify) is typically clearer.

🌟 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

More Questions

Click on any question to see the complete solution with step-by-step explanations

Square of Difference

Solve (x+3)² = (x-3)²: Finding Values When Squared Binomials Are Equal
Click to solve
Solve (x-4)² = (x+2)(x-1): Perfect Square Equals Product
Click to solve