Solve x²-(x+4)²=40: Perfect Square Binomial Challenge

Algebraic Simplification with Perfect Square Expansion

x2−(x+4)2=40 x^2-(x+4)^2=40

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find X
00:03 We'll use shortened multiplication formulas to open the brackets
00:29 Solve the multiplication and square
00:37 Negative times positive is always negative
00:55 Simplify what we can
01:00 Isolate X
01:22 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

x2−(x+4)2=40 x^2-(x+4)^2=40

2

Step-by-step solution

To solve the equation x2−(x+4)2=40 x^2 - (x+4)^2 = 40 , follow these steps:

  • Step 1: Expand the square (x+4)2 (x+4)^2 .
  • Step 2: Substitute the expansion into the original equation.
  • Step 3: Simplify the resulting expression.
  • Step 4: Solve the simplified equation for x x .

Let's work through each step:

Step 1: Expand (x+4)2 (x+4)^2 :
(x+4)2=x2+8x+16(x+4)^2 = x^2 + 8x + 16

Step 2: Substitute this into the original equation:
x2−(x2+8x+16)=40 x^2 - (x^2 + 8x + 16) = 40

Step 3: Simplify the equation:
x2−x2−8x−16=40 x^2 - x^2 - 8x - 16 = 40

Upon simplification, the equation becomes:
−8x−16=40 -8x - 16 = 40

Step 4: Solve for x x :
Add 16 to both sides:
−8x=56 -8x = 56

Divide by −8-8:
x=−568=−7 x = -\frac{56}{8} = -7

Therefore, the solution to the problem is x=−7 x = -7 .

3

Final Answer

x=−7 x=-7

Key Points to Remember

Essential concepts to master this topic
  • Expansion Rule: (x+4)2=x2+8x+16 (x+4)^2 = x^2 + 8x + 16 using perfect square formula
  • Technique: After substitution, x2 x^2 terms cancel leaving −8x−16=40 -8x - 16 = 40
  • Check: Substitute x=−7 x = -7 : (−7)2−(−3)2=49−9=40 (-7)^2 - (-3)^2 = 49 - 9 = 40 ✓

Common Mistakes

Avoid these frequent errors
  • Incorrectly expanding the perfect square binomial
    Don't expand (x+4)2 (x+4)^2 as x2+16 x^2 + 16 = missing the middle term! This gives −16=40 -16 = 40 which is impossible. Always remember the perfect square 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 disappear from the equation?

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Great observation! When you expand and substitute, you get x2−(x2+8x+16)=40 x^2 - (x^2 + 8x + 16) = 40 . The positive and negative x2 x^2 terms cancel each other out, leaving only the linear terms.

How do I remember the perfect square formula?

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Think of it as First + Last + Middle: (x+4)2=x2+42+2(x)(4) (x+4)^2 = x^2 + 4^2 + 2(x)(4) . The middle term is always twice the product of the two terms inside the parentheses.

What if I expanded the left side incorrectly?

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You'd end up with the wrong linear equation! For example, forgetting the middle term 8x 8x would give you −16=40 -16 = 40 , which has no solution. Always double-check your expansion.

Can I solve this using factoring instead?

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Yes! You can rewrite as x2−(x+4)2=40 x^2 - (x+4)^2 = 40 and use the difference of squares formula: (x−(x+4))(x+(x+4))=40 (x-(x+4))(x+(x+4)) = 40 , which gives (−4)(2x+4)=40 (-4)(2x+4) = 40 .

How can I check my answer without substituting?

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Since this simplifies to a linear equation, there should be exactly one solution. If you got multiple answers or no solution, check your algebra steps - especially the perfect square expansion!

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