Solve the Factored Equation: (2+x)(2-x)=0

Q: Why can I set each factor equal to zero?

The Zero Product Property states that if two numbers multiply to give zero, then at least one of them must be zero. Since $ (2+x)(2-x) = 0 $, either $ 2+x = 0 $ or $ 2-x = 0 $!

Q: Do I need to solve both equations separately?

Yes! Each factor gives you a different solution. From $ 2+x = 0 $ you get $ x = -2 $, and from $ 2-x = 0 $ you get $ x = 2 $. Both are correct!

Q: What does ±2 mean in the answer?

The symbol ± means "plus or minus." So $ x = ±2 $ is shorthand for saying both $ x = 2 $ and $ x = -2 $ are solutions.

Q: Is this related to difference of squares?

Absolutely! The expression $ (2+x)(2-x) $ follows the pattern $ (a+b)(a-b) = a^2 - b^2 $. Here it equals $ 4 - x^2 $, which is why both methods give the same answer.

Zero Product Property with Square Differences

Solve:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Let's find what zeroes all parentheses

00:07 Set equal to 0 and isolate X

00:12 Find the two solutions for X

00:17 This is one solution

00:26 And this is the second solution

00:30 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

Solve:

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

Step-by-step solution

We use the abbreviated multiplication formula:

$4-x^2=0$

We isolate the terms and extract the root:

$4=x^2$

$x=\sqrt{4}$

$x=\pm2$

Final Answer

±2

Key Points to Remember

Essential concepts to master this topic

Zero Product Rule: If $ab = 0$ , then $a = 0$ or $b = 0$
Technique: Recognize $(2+x)(2-x) = 4-x^2$ as difference of squares pattern
Check: Substitute both answers: $(2+2)(2-2) = 4 \cdot 0 = 0$ and $(2-2)(2+2) = 0 \cdot 4 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Expanding the product before applying zero product property
Don't expand $(2+x)(2-x)$ to get $4-x^2 = 0$ first = extra work and potential errors! While this method works, it's longer and you might make algebraic mistakes. Always apply the zero product property directly to factored equations.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why can I set each factor equal to zero?

The Zero Product Property states that if two numbers multiply to give zero, then at least one of them must be zero. Since $(2+x)(2-x) = 0$ , either $2+x = 0$ or $2-x = 0$ !

Do I need to solve both equations separately?

Yes! Each factor gives you a different solution. From $2+x = 0$ you get $x = -2$ , and from $2-x = 0$ you get $x = 2$ . Both are correct!

What does ±2 mean in the answer?

The symbol ± means "plus or minus." So $x = ±2$ is shorthand for saying both $x = 2$ and $x = -2$ are solutions.

Is this related to difference of squares?

Absolutely! The expression $(2+x)(2-x)$ follows the pattern $(a+b)(a-b) = a^2 - b^2$ . Here it equals $4 - x^2$ , which is why both methods give the same answer.

Can there be more than two solutions?

For this equation, no. Since we have exactly two factors, we can get at most two solutions. The number of solutions often equals the number of distinct linear factors.

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