There is no possibility of resolving

Simplify the Expression: x² + (-9) Step by Step

Q: Why does x² - 9 = 0 have two solutions?

Because when you square both 3 and -3, you get 9! Since $ 3^2 = 9 $ and $ (-3)^2 = 9 $, both values make the equation true.

Q: How do I recognize a difference of squares?

Look for the pattern $ a^2 - b^2 $! Here, $ x^2 - 9 $ becomes $ x^2 - 3^2 $ since 9 is a perfect square (3²).

Q: What if I can't factor the expression?

Not all expressions are difference of squares! You need two perfect squares separated by subtraction. If factoring doesn't work, try other methods like the quadratic formula.

Q: Why can't I just take the square root of both sides?

You could, but you'd get $ x = ±\sqrt{9} = ±3 $ directly. Factoring shows the steps clearly and helps you understand why there are two solutions!

Q: Do I always write the answer as x = ±3?

Yes! The ± symbol is the standard way to show both positive and negative solutions. It's cleaner than writing $ x = 3 $ or $ x = -3 $ separately.

Quadratic Equations with Difference of Squares

$(+x^2)+(-9)=$

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Understand the problem

$(+x^2)+(-9)=$

Step-by-step solution

To solve the given equation $x^2 - 9 = 0$ , we will utilize the difference of squares technique.

The equation can be written as:

$x^2 - 3^2 = 0$

Recognizing this as a difference of squares, we have:

$(x - 3)(x + 3) = 0$

This implies that either $x - 3 = 0$ or $x + 3 = 0$ .

Solving these two linear equations will give us the values of $x$ .

For $x - 3 = 0$ , adding 3 to both sides gives:

$x = 3$ .

For $x + 3 = 0$ , subtracting 3 from both sides gives:

$x = -3$ .

Therefore, the solutions to the equation $x^2 - 9 = 0$ are $x = 3$ and $x = -3$ .

Conclusively, we have:

The solution to the problem is $\mathbf{x = \pm 3}$ .

Final Answer

$x=±3$

Key Points to Remember

Essential concepts to master this topic

Recognition: Transform $x^2 - 9$ into $x^2 - 3^2$ form
Factoring: Apply difference of squares: $(x-3)(x+3) = 0$
Check: Verify both solutions: $3^2 - 9 = 0$ and $(-3)^2 - 9 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative solution
Don't solve only $x - 3 = 0$ and miss $x + 3 = 0$ = incomplete answer! When you factor $(x-3)(x+3) = 0$ , both factors must equal zero. Always solve both linear equations to get $x = ±3$ .

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why does x² - 9 = 0 have two solutions?

Because when you square both 3 and -3, you get 9! Since $3^2 = 9$ and $(-3)^2 = 9$ , both values make the equation true.

How do I recognize a difference of squares?

Look for the pattern $a^2 - b^2$ ! Here, $x^2 - 9$ becomes $x^2 - 3^2$ since 9 is a perfect square (3²).

What if I can't factor the expression?

Not all expressions are difference of squares! You need two perfect squares separated by subtraction. If factoring doesn't work, try other methods like the quadratic formula.

Why can't I just take the square root of both sides?

You could, but you'd get $x = ±\sqrt{9} = ±3$ directly. Factoring shows the steps clearly and helps you understand why there are two solutions!

Do I always write the answer as x = ±3?

Yes! The ± symbol is the standard way to show both positive and negative solutions. It's cleaner than writing $x = 3$ or $x = -3$ separately.

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