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Bot GPT-4 Turbo To solve the equation \( 9x^2 - 25 = 0 \), you can treat it as a difference of squares, because \( 9x^2 \) is the square of \( 3x \) and \( 25 \) is the square of \( 5 \). The difference of squares formula is:
\[ a^2 - b^2 = (a + b)(a - b) \]
Therefore, \( 9x^2 - 25 \) can be factored as:
\[ 9x^2 - 25 = (3x)^2 - 5^2 = (3x + 5)(3x - 5) \]
Now that we have factored the equation, we set each factor equal to zero and solve for \( x \):
1. \( 3x + 5 = 0 \)
Subtract 5 from both sides to get:
\( 3x = -5 \)
Divide both sides by 3 to get:
\( x = -\frac{5}{3} \)
2. \( 3x - 5 = 0 \)
Add 5 to both sides to get:
\( 3x = 5 \)
Divide both sides by 3 to get:
\( x = \frac{5}{3} \)
The solution set for the equation \( 9x^2 - 25 = 0 \) is \( \{ -\frac{5}{3}, \frac{5}{3} \} \).