Solve the Equation: Finding x in (x+1)² = x² + 13

Quadratic Expansion with Linear Simplification

(x+1)2=x2+13 (x+1)^2=x^2+13

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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 expand the brackets
00:09 Solve the multiplications and squares
00:16 Simplify what we can
00:20 Isolate X
00:38 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(x+1)2=x2+13 (x+1)^2=x^2+13

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 the mentioned formula and expand the parentheses in the expressions in the equation:

(x+1)2=x2+13x2+2⋅x⋅1+12=x2+13x2+2x+1=x2+13 (x+1)^2=x^2+13 \\ x^2+2\cdot x\cdot1+1^2=x^2+13 \\ x^2+2x+1=x^2+13

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+2x+1=x2+132x=12/:2x=6 x^2+2x+1=x^2+13 \\ 2x=12\hspace{8pt}\text{/}:2\\ \boxed{x=6} Therefore, the correct answer is answer B.

3

Final Answer

x=6 x=6

Key Points to Remember

Essential concepts to master this topic
  • Binomial Formula: Expand (x+1)2 (x+1)^2 using a2+2ab+b2 a^2 + 2ab + b^2
  • Technique: Cancel identical terms: x2 x^2 appears on both sides, so subtract it
  • Check: Substitute x = 6: (6+1)2=49 (6+1)^2 = 49 and 62+13=49 6^2 + 13 = 49 ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to expand the binomial correctly
    Don't just write (x+1)2=x2+1 (x+1)^2 = x^2 + 1 = wrong equation! This misses the middle term 2x and leads to impossible solutions. Always use the complete binomial formula: (x+1)2=x2+2x+1 (x+1)^2 = x^2 + 2x + 1 .

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 x^2 term cancel out?

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When you have x2+2x+1=x2+13 x^2 + 2x + 1 = x^2 + 13 , you can subtract x2 x^2 from both sides. This leaves 2x + 1 = 13, which is much simpler to solve!

What if I expanded the binomial wrong?

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If you forget the middle term 2x, you'll get x2+1=x2+13 x^2 + 1 = x^2 + 13 , which gives 1 = 13 - impossible! Always remember: (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2

How do I remember the binomial expansion formula?

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Think of it as "First squared + 2 times First times Second + Second squared". For (x+1)2 (x+1)^2 : x2+2(x)(1)+12 x^2 + 2(x)(1) + 1^2

Can I solve this without expanding?

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You could try taking square roots, but you'd get x+1=±x2+13 x+1 = \pm\sqrt{x^2+13} , which is much harder! Expanding first makes this problem much simpler.

What type of equation is this really?

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Even though it starts with x2 x^2 terms, it's actually a linear equation in disguise! Once the x2 x^2 terms cancel, you're left with 2x = 12.

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