There is no possibility of resolving

Solve (-x)² - 25: Working with Squared Negative Variables

Q: Why does (-x)² equal x² instead of -x²?

The parentheses make all the difference! $(-x)^2$ means you're squaring the entire negative value, and negative times negative equals positive. So $(-x)^2 = x^2$.

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

Look for the pattern $a^2 - b^2$! Here we have $x^2 - 25 = x^2 - 5^2$, which factors as $(x-5)(x+5)$. Both terms are perfect squares separated by subtraction.

Q: Why are there two solutions instead of one?

Because we're solving a quadratic equation! When we factor $(x-5)(x+5) = 0$, either factor can equal zero. This gives us $x = 5$ or $x = -5$.

Q: What if I forgot the difference of squares formula?

You can still solve $x^2 = 25$ by taking the square root of both sides. Just remember: $\sqrt{x^2} = |x|$, so you get $x = ±5$.

Quadratic Equations with Perfect Square Patterns

$(-x)^2-(+25)=$

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Understand the problem

$(-x)^2-(+25)=$

Step-by-step solution

To solve this problem, we'll proceed as follows:

Step 1: Recognize that $(-x)^2 = x^2$ .
Step 2: Establish the equation by substituting: $x^2 - 25 = 0$ .
Step 3: Recognize this as a difference of squares: $(x - 5)(x + 5) = 0$ .
Step 4: Solve for $x$ from the factors: $x - 5 = 0$ and $x + 5 = 0$ .
Step 5: Find solutions: $x = 5$ and $x = -5$ .

Therefore, the solution to the equation is $x = \pm 5$ .

The correct choice among the options provided is $x = \pm 5$ .

Final Answer

$x=±5$

Key Points to Remember

Essential concepts to master this topic

Rule: $(-x)^2 = x^2$ because negative times negative equals positive
Technique: Factor difference of squares: $x^2 - 25 = (x-5)(x+5) = 0$
Check: Substitute both solutions: $(-5)^2 - 25 = 0$ and $(5)^2 - 25 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking (-x)² equals -x²
Don't confuse $(-x)^2$ with $-x^2$ = wrong sign! The parentheses make all the difference: $(-x)^2 = (-1 \cdot x)^2 = (-1)^2 \cdot x^2 = x^2$ . Always remember that squaring a negative gives positive.

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 (-x)² equal x² instead of -x²?

The parentheses make all the difference! $(-x)^2$ means you're squaring the entire negative value, and negative times negative equals positive. So $(-x)^2 = x^2$ .

How do I know this is a difference of squares?

Look for the pattern $a^2 - b^2$ ! Here we have $x^2 - 25 = x^2 - 5^2$ , which factors as $(x-5)(x+5)$ . Both terms are perfect squares separated by subtraction.

Why are there two solutions instead of one?

Because we're solving a quadratic equation! When we factor $(x-5)(x+5) = 0$ , either factor can equal zero. This gives us $x = 5$ or $x = -5$ .

What if I forgot the difference of squares formula?

You can still solve $x^2 = 25$ by taking the square root of both sides. Just remember: $\sqrt{x^2} = |x|$ , so you get $x = ±5$ .

How do I check my answer?

Substitute both solutions back into the original equation. For $x = 5$ : $(-5)^2 - 25 = 25 - 25 = 0$ ✓. For $x = -5$ : $(-(−5))^2 - 25 = 5^2 - 25 = 0$ ✓.

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