Question
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\( x ^ { 2 } - 10 x + 16 = \square \times \square \)

Ask by French Bates. in the United States
Mar 10,2025

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Answer

\( x^2 - 10x + 16 = (x - 2)(x - 8) \)

Solution

1. We start with the quadratic equation: \[ x^2 - 10x + 16 \] 2. We need to factor it into the form: \[ (x + a)(x + b) \] where \( a \) and \( b \) satisfy: - The sum: \( a + b = -10 \) - The product: \( a \times b = 16 \) 3. The numbers that satisfy these conditions are \(-2\) and \(-8\) because: \[ (-2) + (-8) = -10 \quad \text{and} \quad (-2) \times (-8) = 16 \] 4. Therefore, the factorization is: \[ x^2 - 10x + 16 = (x - 2)(x - 8) \]

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Bonus Knowledge

This equation can be factored to find its roots. By rearranging it, we are looking for values of \(x\) that satisfy the quadratic equation \(x^2 - 10x + 16 = 0\). The roots can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = -10\), and \(c = 16\). Plugging these values into the formula will yield the values of \(x\). This type of quadratic problem often shows up in real-life situations, like calculating the trajectory of an object, determining the optimal dimensions for a piece of land, or even in finance for optimizing costs. By recognizing the form of the equation, it becomes easier to apply these concepts in various careers such as engineering, economics, and even architecture!

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