5. e^(2t)-3e^t = 0

검증된 단계별 안내

Rewrite the equation \(e^{2t} - 3e^t = 0\) by recognizing that \(e^{2t}\) can be expressed as \((e^t)^2\). This allows us to treat the equation like a quadratic in terms of \(e^t\).

Let \(x = e^t\). Substitute this into the equation to get \(x^2 - 3x = 0\).

Factor the quadratic equation: \(x(x - 3) = 0\). This gives two possible solutions for \(x\): \(x = 0\) or \(x = 3\).

Recall that \(x = e^t\), and since \(e^t\) is never zero for any real \(t\), discard \(x = 0\). Focus on \(e^t = 3\).

Solve for \(t\) by taking the natural logarithm of both sides: \(t = \ln(3)\). This gives the solution for \(t\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Functions

Exponential functions involve variables in the exponent, such as e^t. Understanding their properties, like how e^(a+b) = e^a * e^b, helps simplify and solve equations involving exponentials.

Substitution Method

The substitution method replaces a complex expression with a simpler variable to transform the equation into a more familiar form, such as turning e^t into x, making it easier to solve.

Solving Quadratic Equations

After substitution, the equation often becomes quadratic. Knowing how to solve quadratic equations using factoring, the quadratic formula, or completing the square is essential to find the variable's values.