Solve (x+3)² = x² + 9: Perfect Square Equation Challenge

Perfect Square Expansion with Algebraic Simplification

Solve for x:

(x+3)2=x2+9 (x+3)^2=x^2+9

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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 shortened multiplication formulas to open the parentheses
00:12 Solve the multiplications and squares
00:19 Simplify what we can
00:25 Isolate X
00:30 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve for x:

(x+3)2=x2+9 (x+3)^2=x^2+9

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 then apply the formula we mentioned and expand the parentheses in the expression in the equation:

(x+3)2=x2+9x2+2⋅x⋅3+32=x2+9x2+6x+9=x2+9 (x+3)^2=x^2+9 \\ x^2+2\cdot x\cdot3+3^2=x^2+9\\ x^2+6x+9=x^2+9 We'll continue and combine like terms, by moving terms around. 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:

x2+6x+9=x2+96x=0/:6x=0 x^2+6x+9=x^2+9 \\ 6x=0\hspace{8pt}\text{/}:6\\ \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
  • Rule: Expand (x+3)2 (x+3)^2 using formula (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2
  • Technique: Subtract identical terms from both sides: x2+6x+9−x2=x2+9−x2 x^2 + 6x + 9 - x^2 = x^2 + 9 - x^2
  • Check: Substitute x = 0: (0+3)2=9 (0+3)^2 = 9 and 02+9=9 0^2 + 9 = 9 ✓

Common Mistakes

Avoid these frequent errors
  • Incorrectly expanding the perfect square
    Don't forget the middle term when expanding (x+3)2 (x+3)^2 = wrong result like x2+9 x^2 + 9 ! This misses the crucial 6x 6x term that makes the equation solvable. Always use the complete formula (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 .

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

\( (x+y)^2 \)

FAQ

Everything you need to know about this question

Why does the x2 x^2 term cancel out?

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Both sides have x2 x^2 terms! When you expand (x+3)2=x2+6x+9 (x+3)^2 = x^2 + 6x + 9 , you can subtract x2 x^2 from both sides, leaving 6x + 9 = 9.

How do I remember the perfect square formula?

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Think "First, Twice, Last": (x+3)2 (x+3)^2 = First squared + Twice the product + Last squared = x2+2(x)(3)+32 x^2 + 2(x)(3) + 3^2 !

What if I expand wrong and get x = 3?

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Check your work! If x=3 x = 3 , then (3+3)2=36 (3+3)^2 = 36 but 32+9=18 3^2 + 9 = 18 . Since 36 ≠ 18, this proves the expansion was incorrect.

Why is this easier than solving a quadratic?

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Because the x2 x^2 terms cancel! This transforms what looks like a quadratic equation into a simple linear equation 6x=0 6x = 0 .

Can I solve this by taking square roots?

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No! The equation isn't (x+3)2=(x)2+9 (x+3)^2 = (x)^2 + 9 in perfect square form. You must expand first to see the true structure of the equation.

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