Algebra

x^2 - 9 = 0

Step-by-step solution with explanation

Final Answer

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Step-by-step solution

1

Recognize this as difference of squares

Notice that x² and 9 are both perfect squares. This matches the pattern a² - b² = 0, which we can factor quickly.
2

Factor using difference of squares

The difference of squares formula says a² - b² = (a - b)(a + b). Here a = x and b = 3, so x² - 9 factors into (x - 3)(x + 3).
3

Set each factor equal to zero

If two things multiply to zero, at least one of them must be zero. This is called the Zero Product Property. We set each factor equal to zero separately.
4

Solve each equation for x

Adding 3 to the first equation gives x = 3. Subtracting 3 from the second gives x = -3. Both values are solutions to the original equation.

Understanding this problem

Learning Insight

The difference of squares pattern works because (a - b)(a + b) always expands to a² + ab - ab - b², and the middle terms cancel out. This cancellation is WHY the shortcut exists — opposite signs create opposite middle terms that wipe each other out.

Quick Tip

Whenever you see x² minus a perfect square (like 1, 4, 9, 16, 25...), immediately think difference of squares. Just take the square root of the number: x² - 9 → √9 = 3 → answers are always +3 and -3.

Common Mistake

Students often find only the positive solution x = 3 and forget x = -3. Remember: squaring a negative number gives a positive result, so negative answers are always valid solutions for x² equations.