Factor: (x + 9)(x – 8) = 0 → x = 8 (positive) - Richter Guitar
Understanding Factor: (x + 9)(x – 8) = 0 → x = 8 (Positive Solution)
Understanding Factor: (x + 9)(x – 8) = 0 → x = 8 (Positive Solution)
When solving equations like (x + 9)(x – 8) = 0, one of the key concepts students and problem solvers encounter is how factoring helps identify solutions. Let’s explore this equation step by step and clarify the outcome — why x = 8 is the correct and positive solution.
Solving (x + 9)(x – 8) = 0
Understanding the Context
This equation is a product of two factors set equal to zero:
(x + 9)(x – 8) = 0
According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero:
- x + 9 = 0 → x = –9
- x – 8 = 0 → x = 8
So the two solutions to the equation are x = –9 and x = 8.
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Key Insights
Why is x = 8 the Positive Solution?
Among the two real solutions—–9 and 8—only x = 8 is positive. To summarize:
- x = –9: A negative number; not relevant if we are only considering positive values.
- x = 8: A positive solution, often highlighted in algebra when restricting to positive roots (useful in real-world applications like measurements, growth modeling, or financial calculations).
How Factorization Simplifies Problem Solving
Factoring transforms a quadratic expression into simpler terms, making it easier to find zeros of the function. By recognizing that (x + 9) and (x – 8) are roots, we instantly identify the values where the expression equals zero—no advanced calculation needed.
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Final Thoughts
Understanding that (x + 9)(x – 8) = 0 leads to x = –9 and x = 8 is essential in algebra. When asked specifically for the positive solution, x = 8 is the correct and relevant answer. Mastering factoring helps build a foundation for solving more complex equations and applying mathematical reasoning in diverse fields.
Key Takeaway:
When solving (x + 9)(x – 8) = 0, while both x = –9 and x = 8 satisfy the equation, only x = 8 is positive — a clear, positive solution valuable in both academic and real-world contexts.
Keywords: algebra equation, factoring quadratic, (x+9)(x-8)=0, positive root, zero product property, solving linear equations, math tutorial, factoring basics, equation solutions, x = 8, mathematical problem solving.
