Solve (x+3)² = (x-3)²: Equal Squared Binomials

Q: Can I solve this without expanding the squares?

Yes! If $ (x+3)^2 = (x-3)^2 $, then taking square roots gives us $ |x+3| = |x-3| $. This means the expressions inside have equal absolute values.

Q: What does it mean geometrically when the squares are equal?

Geometrically, this means x is equidistant from 3 and -3 on the number line. The only point equidistant from 3 and -3 is exactly halfway between them: x = 0.

Q: How do I know I didn't make an algebra mistake?

Always substitute your answer back into the original equation. With x = 0: $ (0+3)^2 = 9 $ and $ (0-3)^2 = 9 $, so both sides equal 9!

Q: Could there be other solutions I'm missing?

No! Once the equation simplifies to $ 12x = 0 $, there's only one solution: x = 0. Linear equations always have exactly one solution (unless there are no solutions or infinitely many).

Squared Binomial Equations with Zero Solutions

Solve the following equation:

$(x+3)^2=(x-3)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Let's use shortened multiplication formulas to open the brackets

00:14 Let's calculate the multiplications

00:24 Let's simplify what we can

00:37 Let's isolate X

00:43 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:

$(x+3)^2=(x-3)^2$

Step-by-step solution

Let's examine the given equation:

$(x+3)^2=(x-3)^2$ First, let's simplify the equation, for this we'll use the perfect square formula for a binomial squared:

$(a\pm b)^2=a^2\pm2ab+b^2$ ,

We'll start by opening the parentheses on both sides simultaneously using the perfect square formula mentioned, then we'll move terms and combine like terms, and in the final step we'll solve the resulting simplified equation:

$(x+3)^2=(x-3)^2 \\ \downarrow\\ x^2+2\cdot x\cdot3+3^2= x^2-2\cdot x\cdot3+3^2 \\ x^2+6x+9= x^2-6x+9 \\ x^2+6x+9- x^2+6x-9 =0\\ 12x=0\hspace{6pt}\text{/}:12\\ \boxed{x=0}$ Therefore, the correct answer is answer A.

Final Answer

$x=0$

Key Points to Remember

Essential concepts to master this topic

Rule: Equal squared expressions mean the bases differ by zero
Technique: Expand both sides: $x^2+6x+9 = x^2-6x+9$
Check: Substitute x=0: $(0+3)^2 = (0-3)^2$ gives 9=9 ✓

Common Mistakes

Avoid these frequent errors

Assuming two solutions exist because of squared terms
Don't think every squared equation has two solutions = missing the cancellation! When terms cancel perfectly, only one solution remains. Always expand completely and combine like terms to see the true 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 is there only one solution when we have squared terms?

Even though we have squares, the x² terms cancel out when we expand! After expanding $(x+3)^2 = (x-3)^2$ , we get a simple linear equation: $12x = 0$ .

Can I solve this without expanding the squares?

Yes! If $(x+3)^2 = (x-3)^2$ , then taking square roots gives us $|x+3| = |x-3|$ . This means the expressions inside have equal absolute values.

What does it mean geometrically when the squares are equal?

Geometrically, this means x is equidistant from 3 and -3 on the number line. The only point equidistant from 3 and -3 is exactly halfway between them: x = 0.

How do I know I didn't make an algebra mistake?

Always substitute your answer back into the original equation. With x = 0: $(0+3)^2 = 9$ and $(0-3)^2 = 9$ , so both sides equal 9!

Could there be other solutions I'm missing?

No! Once the equation simplifies to $12x = 0$ , there's only one solution: x = 0. Linear equations always have exactly one solution (unless there are no solutions or infinitely many).

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Solve (x+3)² = (x-3)²: Equal Squared Binomials

Squared Binomial Equations with Zero Solutions

❤️ 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 is there only one solution when we have squared terms?

Can I solve this without expanding the squares?

What does it mean geometrically when the squares are equal?

How do I know I didn't make an algebra mistake?

Could there be other solutions I'm missing?

🌟 Unlock Your Math Potential

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