Solve the Equation: (x-1)² = x² | Comparing Perfect Squares

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

While taking square roots works for simple equations like $ x^2 = 9 $, here you'd get x - 1 = ±x, which leads to contradictory solutions. It's safer to expand first using the perfect square formula.

Q: How do I remember the perfect square formula?

Think of it as "First squared, minus twice the product, plus second squared": $ (a-b)^2 = a^2 - 2ab + b^2 $. Practice with simple examples like $ (x-3)^2 $!

Q: What if I expand incorrectly?

Double-check by using FOIL: $ (x-1)(x-1) = x^2 - x - x + 1 = x^2 - 2x + 1 $. This gives the same result as the perfect square formula.

Q: Why does this equation have only one solution instead of two?

After expanding and simplifying, we get -2x = -1, which is a linear equation with exactly one solution. The original quadratic terms cancelled out!

Q: How can I verify my answer is correct?

Substitute $ x = \frac{1}{2} $ back: Left side = $ (\frac{1}{2}-1)^2 = (-\frac{1}{2})^2 = \frac{1}{4} $, Right side = $ (\frac{1}{2})^2 = \frac{1}{4} $. Both sides equal!

Quadratic Equations with Perfect Square Expansion

Solve the following equation:

$(x-1)^2=x^2$

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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 the shortened multiplication formulas to expand the brackets

00:10 Calculate the multiplications and the square

00:17 Simplify what we can

00:24 Isolate X

00:35 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$(x-1)^2=x^2$

Step-by-step solution

Let's examine the given equation:

$(x-1)^2=x^2$ First, let's simplify the equation, using 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 resulting simplified equation:

$(x-1)^2=x^2 \\ \downarrow\\ x^2-2\cdot x\cdot1+1^2=x^2\\ x^2-2x+1= x^2\\ -2x=-1\hspace{6pt}\text{/}:(-2)\\ \boxed{x=\frac{1}{2}}$

Therefore, the correct answer is answer A.

Final Answer

$x=\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Formula: Apply $(a-b)^2 = a^2 - 2ab + b^2$ to expand perfect squares
Technique: Expand $(x-1)^2 = x^2 - 2x + 1$ , then subtract $x^2$ from both sides
Check: Substitute $x = \frac{1}{2}$ : $(\frac{1}{2}-1)^2 = (\frac{1}{2})^2$ gives $\frac{1}{4} = \frac{1}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of both sides immediately
Don't take √ of both sides to get x-1 = ±x = wrong answers like x = -1/2! This ignores the squared terms that need expanding first. Always expand the perfect square using the binomial formula, then solve 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?

While taking square roots works for simple equations like $x^2 = 9$ , here you'd get x - 1 = ±x, which leads to contradictory solutions. It's safer to expand first using the perfect square formula.

How do I remember the perfect square formula?

Think of it as "First squared, minus twice the product, plus second squared": $(a-b)^2 = a^2 - 2ab + b^2$ . Practice with simple examples like $(x-3)^2$ !

What if I expand incorrectly?

Double-check by using FOIL: $(x-1)(x-1) = x^2 - x - x + 1 = x^2 - 2x + 1$ . This gives the same result as the perfect square formula.

Why does this equation have only one solution instead of two?

After expanding and simplifying, we get -2x = -1, which is a linear equation with exactly one solution. The original quadratic terms cancelled out!

How can I verify my answer is correct?

Substitute $x = \frac{1}{2}$ back: Left side = $(\frac{1}{2}-1)^2 = (-\frac{1}{2})^2 = \frac{1}{4}$ , Right side = $(\frac{1}{2})^2 = \frac{1}{4}$ . Both sides equal!

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Solve the Equation: (x-1)² = x² | Comparing Perfect Squares

Quadratic Equations with Perfect Square Expansion

❤️ 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

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

How do I remember the perfect square formula?

What if I expand incorrectly?

Why does this equation have only one solution instead of two?

How can I verify my answer is correct?

🌟 Unlock Your Math Potential

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