Solve for X: Finding the Value in (x+3)² = x² + 15

Perfect Square Binomial with Linear Solutions

What is the value of x?

(x+3)2=x2+15 (x+3)^2=x^2+15

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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 abbreviated multiplication formulas to open the brackets
00:10 Solve the multiplications and squares
00:18 Simplify what we can
00:23 Isolate X
00:39 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

What is the value of x?

(x+3)2=x2+15 (x+3)^2=x^2+15

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

(x+3)2=x2+15x2+2x3+32=x2+15x2+6x+9=x2+15 (x+3)^2=x^2+15 \\ x^2+2\cdot x\cdot3+3^2=x^2+15\\ x^2+6x+9=x^2+15 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=x2+156x=6/:6x=1 x^2+6x+9=x^2+15 \\ 6x=6\hspace{8pt}\text{/}:6\\ \boxed{x=1} Therefore, the correct answer is answer B.

3

Final Answer

x=1 x=1

Key Points to Remember

Essential concepts to master this topic
  • Binomial Formula: (a+b)^2 = a^2 + 2ab + b^2 for expansion
  • Technique: Expand (x+3)^2 = x^2 + 6x + 9 then cancel like terms
  • Check: Substitute x=1: (1+3)^2 = 16 and 1^2 + 15 = 16

Common Mistakes

Avoid these frequent errors
  • Incorrectly expanding the perfect square
    Don't expand (x+3)^2 as x^2 + 9 = missing the middle term! This gives 6x = 6 instead of the correct expansion. Always remember the binomial formula includes the 2ab term: (x+3)^2 = x^2 + 6x + 9.

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 x^2 term disappear when solving?

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When you expand both sides, you get x^2 + 6x + 9 = x^2 + 15. The x^2 terms are identical on both sides, so they cancel out when you subtract, leaving just 6x + 9 = 15.

What's the middle term in (x+3)^2?

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The middle term is 2ab from the formula (a+b)^2 = a^2 + 2ab + b^2. Here, a = x and b = 3, so the middle term is 2 \cdot x \cdot 3 = 6x.

Can I solve this without expanding the binomial?

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Not easily! You could try taking square roots of both sides, but that gets complicated with the mixed terms. Expanding using the binomial formula is the most straightforward approach.

How do I remember the binomial expansion formula?

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Think "First, Outer, Inner, Last" or remember the pattern:

  • Square the first term
  • Add twice the product of both terms
  • Add the square of the second term

Why isn't the answer x = 3?

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Let's check: (3+3)^2 = 36 but 3^2 + 15 = 24. Since 36 ≠ 24, x = 3 doesn't work. Always verify your answer!

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