Solve the Quadratic Puzzle: Factoring x² - 64

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

Look for the pattern $ a^2 - b^2 $! Both terms must be perfect squares separated by a minus sign. Here, $ x^2 $ and $ 64 = 8^2 $ are both perfect squares.

Q: What if I can't remember which perfect square 64 is?

Try counting up: $ 1^2 = 1, 2^2 = 4, 3^2 = 9, ..., 8^2 = 64 $. Or work backwards: what number times itself gives 64? Practice memorizing squares 1-12!

Q: Why does the order matter in the factored form?

It doesn't! $ (x - 8)(x + 8) $ equals $ (x + 8)(x - 8) $. But the question format $ (x - \_)(\_+ x) $ shows you exactly where to put the 8.

Q: Can I use FOIL to check my answer?

Absolutely! Expand $ (x - 8)(x + 8) $: First: $ x \cdot x = x^2 $, Outer: $ x \cdot 8 = 8x $, Inner: $ -8 \cdot x = -8x $, Last: $ -8 \cdot 8 = -64 $. Result: $ x^2 + 8x - 8x - 64 = x^2 - 64 $ ✓

Q: What's the easiest way to remember the difference of squares formula?

Think: "square minus square equals difference times sum". The factors are always $ (a - b) $ and $ (a + b) $ - one subtracts, one adds!

Difference of Squares with Perfect Square Recognition

Fill in the missing element to obtain a true expression:

$x^2-64=(x-_—)(_—+x)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing value

00:13 We'll use the commutative law

00:21 We'll use the abbreviated multiplication formulas

00:37 We'll substitute 64 as B squared

00:47 We'll extract the root to find B

00:52 B is the missing element

00:57 And this is the solution to the question

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:

$x^2-64=(x-_—)(_—+x)$

Step-by-step solution

To solve this problem, we need to recognize the expression $x^2 - 64$ as a difference of squares.

The difference of squares formula states: $a^2 - b^2 = (a - b)(a + b)$ .

In this problem, we identify that:

$a = x$
$b^2 = 64$ , which means $b = \sqrt{64} = 8$

Therefore, applying the formula gives us:

$x^2 - 64 = (x - 8)(x + 8)$

This indicates that the missing element in the expression $(x - \_)(\_ + x)$ is $8$ .

Thus, the correct answer to fill in the missing element is $\boxed{8}$ , corresponding to choice 4.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: $a^2 - b^2 = (a - b)(a + b)$ for difference of squares
Recognition: Identify $64 = 8^2$ so $x^2 - 64 = x^2 - 8^2$
Verification: Expand $(x - 8)(x + 8) = x^2 - 64$ using FOIL ✓

Common Mistakes

Avoid these frequent errors

Confusing the square root of the constant term
Don't think $\sqrt{64} = 6$ or another wrong value = incorrect factorization! Students often miscalculate basic square roots or confuse 64 with other perfect squares. Always double-check: $8 \times 8 = 64$ , so $\sqrt{64} = 8$ .

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 if something is a difference of squares?

Look for the pattern $a^2 - b^2$ ! Both terms must be perfect squares separated by a minus sign. Here, $x^2$ and $64 = 8^2$ are both perfect squares.

What if I can't remember which perfect square 64 is?

Try counting up: $1^2 = 1, 2^2 = 4, 3^2 = 9, ..., 8^2 = 64$ . Or work backwards: what number times itself gives 64? Practice memorizing squares 1-12!

Why does the order matter in the factored form?

It doesn't! $(x - 8)(x + 8)$ equals $(x + 8)(x - 8)$ . But the question format $(x - \_)(\_+ x)$ shows you exactly where to put the 8.

Can I use FOIL to check my answer?

Absolutely! Expand $(x - 8)(x + 8)$ : First: $x \cdot x = x^2$ , Outer: $x \cdot 8 = 8x$ , Inner: $-8 \cdot x = -8x$ , Last: $-8 \cdot 8 = -64$ . Result: $x^2 + 8x - 8x - 64 = x^2 - 64$ ✓

What's the easiest way to remember the difference of squares formula?

Think: "square minus square equals difference times sum". The factors are always $(a - b)$ and $(a + b)$ - one subtracts, one adds!

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Solve the Quadratic Puzzle: Factoring x² - 64

Difference of Squares with Perfect Square Recognition

❤️ 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 if something is a difference of squares?

What if I can't remember which perfect square 64 is?

Why does the order matter in the factored form?

Can I use FOIL to check my answer?

What's the easiest way to remember the difference of squares formula?

🌟 Unlock Your Math Potential

More Questions

Difference of squares

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

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

Fill the Blanks: Solve the Quadratic Equation (_+3)·(_-3) = x²-9

Complete the Expression: (x+)(x-) = x²-121

Solve the Quadratic Puzzle: Factoring x² - 64

Difference of Squares with Perfect Square Recognition

❤️ 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 if something is a difference of squares?

What if I can't remember which perfect square 64 is?

Why does the order matter in the factored form?

Can I use FOIL to check my answer?

What's the easiest way to remember the difference of squares formula?

🌟 Unlock Your Math Potential

More Questions

Difference of squares

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

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

Fill the Blanks: Solve the Quadratic Equation (_+3)·(_-3) = x²-9

Complete the Expression: (x+__)(x-__) = x²-121

Complete the Expression: (x+)(x-) = x²-121