Completing the Polynomial: Fill in the Gap in 2x^2 - __ = 2(x-4)(x+4)

Difference of Squares with Missing Terms

Fill in the missing element to obtain a true expression:

2x2−—=2(x−4)⋅(x+4) 2x^2-_{_—}=2(x-4)\cdot(x+4)

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1

Understand the problem

Fill in the missing element to obtain a true expression:

2x2−—=2(x−4)⋅(x+4) 2x^2-_{_—}=2(x-4)\cdot(x+4)

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Step 1: Expand the expression on the right side of the equation.
  • Step 2: Compare it with the left-hand side equation and find the missing number.

Now, let's work through each step:
Step 1: Expand the expression 2(x−4)(x+4)2(x-4)(x+4). Using the difference of squares, this becomes:

2(x2−42)=2(x2−16)=2x2−32 2(x^2 - 4^2) = 2(x^2 - 16) = 2x^2 - 32

Step 2: Compare it with the original left side 2x2−_=2x2−322x^2 - \_ = 2x^2 - 32.

The missing number must be 32 so that both sides of the equation are equal.

Therefore, the solution to the problem is 32\textbf{32}.

3

Final Answer

32

Key Points to Remember

Essential concepts to master this topic
  • Pattern Recognition: Identify (x−a)(x+a)=x2−a2 (x-a)(x+a) = x^2 - a^2 structure
  • Expand Method: Calculate 2(x−4)(x+4)=2(x2−16)=2x2−32 2(x-4)(x+4) = 2(x^2-16) = 2x^2-32
  • Verification: Check that 2x2−32=2(x−4)(x+4) 2x^2-32 = 2(x-4)(x+4) when expanded ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to distribute the coefficient outside parentheses
    Don't expand just (x-4)(x+4) = x²-16 and forget the 2! This gives the wrong missing term of 16 instead of 32. Always multiply the entire difference of squares result by any coefficient outside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

Solve:

\( (2+x)(2-x)=0 \)

FAQ

Everything you need to know about this question

What is the difference of squares pattern?

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The difference of squares is (x−a)(x+a)=x2−a2 (x-a)(x+a) = x^2 - a^2 . When you multiply conjugate binomials (same terms, opposite signs), the middle terms cancel out, leaving only the difference of the squared terms.

Why can't I just guess the missing number?

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Guessing might work sometimes, but algebraic expansion gives you the exact answer every time. Plus, understanding the process helps you solve similar problems with different numbers or variables.

Do I always get a difference of squares from (x-a)(x+a)?

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Yes! This pattern always produces x2−a2 x^2 - a^2 because the middle terms +ax +ax and −ax -ax cancel out when you use FOIL.

What if there's a coefficient in front like the 2 in this problem?

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Always distribute that coefficient to every term inside! Here, 2(x2−16) 2(x^2-16) becomes 2x2−32 2x^2-32 , not 2x2−16 2x^2-16 .

How can I check my answer is correct?

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Substitute your missing number back into the original equation and expand both sides. If 2x2−32=2(x−4)(x+4) 2x^2-32 = 2(x-4)(x+4) when you expand the right side, you're correct!

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