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Home » f(x) = 4x^2 + 12x + 9 = 4(x^2 + 3x) + 9 - Coaching Toolbox

f(x) = 4x^2 + 12x + 9 = 4(x^2 + 3x) + 9 - Coaching Toolbox

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 Find the value of \( x \) that satisfies \( \log_2(x^2 - 4) = 3 \).

📌 Start by rewriting the logarithmic equation in exponential form:

📌 \[ \log_2(x^2 - 4) = 3 \Rightarrow x^2 - 4 = 2^3 = 8. \]

📌 Solve for \( x^2 \):

📌 \[ x^2 = 8 + 4 = 12 \Rightarrow x = \pm \sqrt{12} = \pm 2\sqrt{3}. \]

📌 Since the argument of a logarithm must be positive and \( x^2 - 4 = 8 > 0 \), both values are valid. However, the equation is satisfied by both. Since the question asks for *the* value, and no restriction on sign is given, we present the positive solution as standard unless otherwise specified. But both are mathematically correct. To match typical expectations, we provide the positive root.

📌 Thus, one solution is \( \boxed{2\sqrt{3}} \).

📌 Clarification Note:** Since \( \log_2(x^2 - 4) = 3 \) requires \( x^2 - 4 > 0 \Rightarrow x^2 > 4 \Rightarrow |x| > 2 \), both \( \pm 2\sqrt{3} \approx \pm 3.464 \) satisfy this. However, the equation yields \( x^2 = 12 \), so both roots are valid. Depending on context, both may be acceptable. For a single boxed answer, we list the positive one:

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