Completing the Expression: Find the Missing Numbers in x^2-36=(x-)(+x)

Q: How do I know this is a difference of squares?

Look for the pattern $ a^2 - b^2 $ where both terms are perfect squares separated by subtraction. Here, $ x^2 $ and $ 36 = 6^2 $ fit this pattern perfectly!

Q: Why is the answer 6 and not 36?

Because $ 36 = 6^2 $, so we need the square root of 36, which is 6. The difference of squares formula uses $ a $ and $ b $, not $ a^2 $ and $ b^2 $ in the factored form.

Q: What if I can't recognize perfect squares quickly?

Practice common perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. For larger numbers, try taking the square root - if you get a whole number, it's a perfect square!

Q: How do I check my factorization is correct?

Use FOIL to expand your answer: $ (x-6)(x+6) = x^2 + 6x - 6x - 36 = x^2 - 36 $. The middle terms cancel out, giving you back the original expression!

Q: Does the order of factors matter?

No! $ (x-6)(x+6) $ and $ (x+6)(x-6) $ are equivalent due to the commutative property of multiplication. Both expand to the same expression.

Difference of Squares with Perfect Square Constants

Fill in the missing element to obtain a true expression:

$x^2-36=(x-_—)\cdot(_—+x)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the missing element to obtain a true expression:

$x^2-36=(x-_—)\cdot(_—+x)$

Step-by-step solution

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

Step 1: Identify the given expression and recognize the form.
Step 2: Apply the difference of squares formula.
Step 3: Determine the missing element.
Step 4: Verify the solution against the possible choices.

Now, let's work through each step:
Step 1: The given expression is $x^2 - 36$ . This resembles a difference of squares, which is $a^2 - b^2$ .
Step 2: Recognize that $x^2$ represents $a^2$ and $36$ represents $b^2$ .
Step 3: Find $b$ such that $b^2 = 36$ . This gives $b = 6$ because $6^2 = 36$ .
Step 4: The difference of squares formula states $a^2 - b^2 = (a - b)(a + b)$ . So we rewrite $x^2 - 36$ as $(x - 6)(x + 6)$ .

Therefore, the missing element that makes the expression true is $6$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Identify $a^2 - b^2$ form where both terms are perfect squares
Formula Application: Use $a^2 - b^2 = (a-b)(a+b)$ with $x^2 - 36 = (x-6)(x+6)$
Verification Check: Expand $(x-6)(x+6) = x^2 - 36$ to confirm correctness ✓

Common Mistakes

Avoid these frequent errors

Finding square root of coefficient incorrectly
Don't assume the missing number is 36 because that's what you see = $(x-36)(36+x)$ gives $x^2 - 1296$ ! You need the square root of 36, not 36 itself. Always find $\sqrt{b^2}$ to get the correct factorization.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

How do I know this is a difference of squares?

Look for the pattern $a^2 - b^2$ where both terms are perfect squares separated by subtraction. Here, $x^2$ and $36 = 6^2$ fit this pattern perfectly!

Why is the answer 6 and not 36?

Because $36 = 6^2$ , so we need the square root of 36, which is 6. The difference of squares formula uses $a$ and $b$ , not $a^2$ and $b^2$ in the factored form.

What if I can't recognize perfect squares quickly?

Practice common perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. For larger numbers, try taking the square root - if you get a whole number, it's a perfect square!

How do I check my factorization is correct?

Use FOIL to expand your answer: $(x-6)(x+6) = x^2 + 6x - 6x - 36 = x^2 - 36$ . The middle terms cancel out, giving you back the original expression!

Does the order of factors matter?

No! $(x-6)(x+6)$ and $(x+6)(x-6)$ are equivalent due to the commutative property of multiplication. Both expand to the same expression.

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Completing the Expression: Find the Missing Numbers in x^2-36=(x-)(+x)

Difference of Squares with Perfect Square Constants

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know this is a difference of squares?

Why is the answer 6 and not 36?

What if I can't recognize perfect squares quickly?

How do I check my factorization is correct?

Does the order of factors matter?

🌟 Unlock Your Math Potential

More Questions

Difference of squares

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

Determine the Validity: Is x^2 - 81 Over (x-9)(x+9) Equal to 1?

Factoring Challenge: Solve the Quadratic x²-6 as a Binomial Product

Factoring a Difference of Squares: Solve x^2 - 49 = (x-_—)(x+_—)

Completing the Expression: Find the Missing Numbers in x^2-36=(x-__)(__+x)

Difference of Squares with Perfect Square Constants

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know this is a difference of squares?

Why is the answer 6 and not 36?

What if I can't recognize perfect squares quickly?

How do I check my factorization is correct?

Does the order of factors matter?

🌟 Unlock Your Math Potential

More Questions

Difference of squares

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

Determine the Validity: Is x^2 - 81 Over (x-9)(x+9) Equal to 1?

Factoring Challenge: Solve the Quadratic x²-6 as a Binomial Product

Factoring a Difference of Squares: Solve x^2 - 49 = (x-_—)(x+_—)

Completing the Expression: Find the Missing Numbers in x^2-36=(x-)(+x)