To solve the quadratic equation \( x^2 + 5x + 6 = 0 \), we can use various methods such as factoring, using the quadratic formula, or completing the square. Here, I'll demonstrate the factoring method since this equation factors easily.
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Factor the quadratic: We need to find two numbers that multiply to the constant term (6) and add up to the coefficient of the \( x \) term (5). The numbers that satisfy these conditions are 2 and 3.
Thus, we can factor the equation as: \[ (x + 2)(x + 3) = 0 \]
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Set each factor to zero: Now, we can set each factor equal to zero to find the solutions for \( x \): \[ x + 2 = 0 \quad \text{or} \quad x + 3 = 0 \]
Solving these equations gives: \[ x = -2 \quad \text{and} \quad x = -3 \]
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Write the solutions: The solutions to the equation \( x^2 + 5x + 6 = 0 \) are: \[ x = -2 \quad \text{and} \quad x = -3 \]
Therefore, the final answer is \( x = -2 \) and \( x = -3 \).