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23) \( a^{2}-6 a-27 \)

Ask by Harris Martin. in the United States
Mar 11,2025

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Tutor-Verified Answer

Answer

\( a^{2}-6a-27 = (a-9)(a+3) \)

Solution

1. We need to factor the quadratic expression \[ a^{2}-6a-27. \] 2. We look for two numbers, \( m \) and \( n \), such that \[ m \times n = -27 \quad \text{and} \quad m + n = -6. \] 3. Consider \( m = -9 \) and \( n = 3 \), since \[ (-9) \times 3 = -27 \quad \text{and} \quad (-9) + 3 = -6. \] 4. Therefore, the factored form is \[ a^{2}-6a-27 = (a-9)(a+3). \]

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The Deep Dive

To solve the quadratic equation \( a^{2}-6a-27 = 0 \), we can use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -6 \), and \( c = -27 \). Plugging in these values gives us \( a = \frac{6 \pm \sqrt{(-6)^{2} - 4(1)(-27)}}{2(1)} \), leading to \( a = \frac{6 \pm \sqrt{36 + 108}}{2} \). Simplifying further will lead you to the solutions of this equation. This quadratic can represent various real-world scenarios, from physics problems involving projectile motion to economics problems analyzing profit and loss. Understanding how to graph this equation can help you identify the roots easily. Graphing reveals where the curve intersects the x-axis, which directly corresponds to the solutions of the equation—those critical points where the value of \( a \) results in zero for the quadratic function.

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