Solve (x+3)² = (x-3)²: Finding Values When Squared Binomials Are Equal

Quadratic Equations with Equal Squared Binomials

(x+3)2=(x3)2 (x+3)^2=(x-3)^2

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:06 Let's find the value of X.
00:09 Use the short multiplication formulas to open all brackets. Ready? Here we go!
00:22 Now, solve the multiplications and calculate the squares. You're doing great!
00:33 Next, simplify everything we're able to. Keep it up!
00:46 Now, isolate X on one side of the equation.
00:55 And that's how we find the solution to the problem! Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(x+3)2=(x3)2 (x+3)^2=(x-3)^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+3)2=(x3)2x2+2x3+32=x22x3+32x2+6x+9=x26x+9 (x+3)^2=(x-3)^2 \\ x^2+2\cdot x\cdot3+3^2=x^2-2\cdot x\cdot3+3^2 \\ x^2+6x+9=x^2-6x+9 \\ 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+6x+9=x26x+912x=0/:12x=0 x^2+6x+9=x^2-6x+9 \\ 12x=0\hspace{8pt}\text{/}:12\\ \boxed{x=0} Therefore, the correct answer is answer A.

3

Final Answer

x=0 x=0

Key Points to Remember

Essential concepts to master this topic
  • Perfect Square Formula: Use (a±b)2=a2±2ab+b2 (a\pm b)^2 = a^2 \pm 2ab + b^2 to expand
  • Technique: Expand both sides: x2+6x+9=x26x+9 x^2 + 6x + 9 = x^2 - 6x + 9
  • Check: Substitute x = 0: (0+3)2=9 (0+3)^2 = 9 and (03)2=9 (0-3)^2 = 9

Common Mistakes

Avoid these frequent errors
  • Taking square roots of both sides immediately
    Don't take square roots of both sides to get x+3 = x-3 = impossible equation! This creates false contradictions and misses the real solution. Always expand the squared binomials first using the perfect square formula.

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?

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Taking square roots would give you x+3=±(x3) x+3 = \pm(x-3) , which creates two cases. But this approach misses the algebraic structure. Expanding first reveals the linear relationship more clearly.

How do I know when squared terms will cancel out?

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When both sides have the same squared expression (like x2 x^2 here), they'll cancel during simplification. This transforms a quadratic-looking equation into a simpler linear one!

What if the constant terms were different?

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If you had (x+3)2=(x2)2 (x+3)^2 = (x-2)^2 , the constants wouldn't cancel. You'd still expand both sides, but you'd get a different linear equation to solve.

Can this type of equation have multiple solutions?

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Generally no! After expanding and simplifying, you usually get a linear equation with exactly one solution, like 12x=0 12x = 0 giving x=0 x = 0 .

How do I remember the perfect square formula?

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Think of FOIL: First, Outer, Inner, Last terms. For (x+3)2 (x+3)^2 , you get x2+3x+3x+9=x2+6x+9 x^2 + 3x + 3x + 9 = x^2 + 6x + 9 . The middle term is always 2ab 2ab !

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