Solve the Equation: (x+1)² = x² | Perfect Square Comparison

Q: Why does the x² term disappear when solving?

After expanding (x+1)² to x² + 2x + 1, you have x² on both sides of the equation. When you subtract x² from both sides, they cancel out, leaving just 2x + 1 = 0.

Q: How do I remember the perfect square formula?

Think FOIL: (x+1)(x+1) = x² + x + x + 1 = x² + 2x + 1. The middle term is always twice the product of the two terms inside the parentheses.

Q: Why is this equation only linear, not quadratic?

Even though we start with x² terms, they cancel out during solving! This leaves us with just 2x + 1 = 0, which is linear (first-degree).

Perfect Square Expansion with Term Cancellation

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Use the shortened multiplication formulas to open the parentheses

00:13 Simplify what we can

00:21 Isolate X

00:29 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

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

Step-by-step solution

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

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

$(x+1)^2=x^2 \\ x^2+2\cdot x\cdot1+1^2=x^2 \\ x^2+2x+1=x^2 \\$ We'll continue and combine like terms, by moving terms between sides. Later - 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:

$x^2+2x+1=x^2 \\ 2x=-1\hspace{8pt}\text{/}:2\\ \boxed{x=-\frac{1}{2}}$ Therefore, the correct answer is answer A.

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Binomial Formula: Apply (a+b)² = a² + 2ab + b² expansion rule
Technique: (x+1)² becomes x² + 2x + 1 before simplifying
Check: Substitute x = -1/2: (-1/2+1)² = (1/2)² = 1/4 = (-1/2)² ✓

Common Mistakes

Avoid these frequent errors

Not expanding the perfect square completely
Don't leave (x+1)² unexpanded and try to solve directly = impossible equation! You miss the crucial 2x term that makes the equation solvable. Always expand using the binomial formula first.

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 term disappear when solving?

After expanding (x+1)² to x² + 2x + 1, you have x² on both sides of the equation. When you subtract x² from both sides, they cancel out, leaving just 2x + 1 = 0.

How do I remember the perfect square formula?

Think FOIL: (x+1)(x+1) = x² + x + x + 1 = x² + 2x + 1. The middle term is always twice the product of the two terms inside the parentheses.

What if I get a positive fraction instead?

Double-check your algebra! The correct expansion gives 2x + 1 = 0, so 2x = -1, making x = -1/2 (negative). A positive answer means an error in signs.

Can I solve this without expanding?

Not really! You need to see what's inside the parentheses. Taking square roots would give |x+1| = |x|, which creates a more complex absolute value equation.

Why is this equation only linear, not quadratic?

Even though we start with x² terms, they cancel out during solving! This leaves us with just 2x + 1 = 0, which is linear (first-degree).

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