Question
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Tutor-Verified Answer
Answer
The solutions are \( x = 5 \) and \( x = -5 \).
Solution
1. Start with the equation:
\[
x^2 - 25 = 0
\]
2. Recognize that the equation is a difference of two squares. It can be factored as:
\[
x^2 - 25 = (x - 5)(x + 5)
\]
3. Set each factor equal to zero:
\[
x - 5 = 0 \quad \text{or} \quad x + 5 = 0
\]
4. Solve each equation:
\[
x - 5 = 0 \quad \Longrightarrow \quad x = 5
\]
\[
x + 5 = 0 \quad \Longrightarrow \quad x = -5
\]
5. Therefore, the solutions to the equation are:
\[
x = 5 \quad \text{and} \quad x = -5
\]
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Simplify this solution The Deep Dive
Did you know that the equation \( x^2 - 25 = 0 \) can be solved using the difference of squares? This mathematical method is like splitting a giant chocolate bar in half! You can rewrite the equation as \( (x - 5)(x + 5) = 0 \). This means that either \( x - 5 = 0 \) or \( x + 5 = 0 \), giving you the solutions \( x = 5 \) and \( x = -5 \). When it comes to problem-solving, one common mistake is forgetting to consider both factors in the difference of squares! It's easy to think only of the positive root, but always remember that quadratic equations can have two solutions. So, don't skip that extra step—check for both possibilities like a detective looking for clues!
