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What are the zeros of the equation \( y+9 x=x^{3} \) ? \( (3,0) \) \( (0,-3) \) \( (0,0),(3,0) \) and \( (-3,0) \) \( (0,3),(-3,0) \) and \( (9,0) \) \( (0,0),(0,9) \) and \( (0,-9) \)

Ask by Martinez Graham. in the United States
Jan 20,2025

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Answer

The zeros of the equation \( y + 9x = x^{3} \) are \( (0,0) \), \( (3,0) \), and \( (-3,0) \).

Solution

To find the zeros of the equation \( y + 9x = x^{3} \), we look for the points where \( y = 0 \) (i.e., the x-intercepts). 1. **Set \( y = 0 \):** \[ 0 + 9x = x^{3} \] \[ x^{3} - 9x = 0 \] 2. **Factor the equation:** \[ x(x^{2} - 9) = 0 \] \[ x(x - 3)(x + 3) = 0 \] 3. **Solve for \( x \):** \[ x = 0, \quad x = 3, \quad x = -3 \] 4. **Identify the corresponding points:** \[ (0, 0), \quad (3, 0), \quad (-3, 0) \] **Therefore, the zeros of the equation are** \( (0,0), (3,0) \) **and** \( (-3,0) \). **Answer:** \( (0,0),(3,0) \) and \( (-3,0) \)

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The Deep Dive

To find the zeros of the equation \( y + 9x = x^3 \), we first set \( y = 0 \). This gives us the equation \( 0 + 9x = x^3 \), or \( x^3 - 9x = 0 \). Factoring out \( x \), we have \( x(x^2 - 9) = 0 \), which further factors to \( x(x - 3)(x + 3) = 0 \). Thus, the zeros are \( x = 0, 3, -3 \), corresponding to the points \( (0,0), (3,0), (-3,0) \). Now, picture this: each zero you found represents a point where the graph of the function touches or crosses the x-axis. It's like having three cheerful friends (the zeros) joining a party at the x-axis, each bringing their own vibe to the equation! For a bit of further exploration, why not play around with graphing the original equation? Visualizing it can provide a clearer understanding of how those zeros interact with the curve. You might find that the curvature changes around these points, leading to discussions about local maxima and minima!

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