Solve by Graphing x^3=x^2-7

Problem

x^3=x^2−7

Solution

1. Define the functions to be graphed by treating each side of the equation as a separate function.

(y_1)=x^3

(y_2)=x^2−7

2. Analyze the behavior of the functions. The cubic function (y_1) increases rapidly for positive x and decreases for negative x while the quadratic function (y_2) is a parabola opening upwards with a vertex at (0,−7)

3. Identify the intersection point by plotting the graphs. By observing the graphs, the curves intersect at a single point where the xcoordinate satisfies the equation.

4. Estimate the xvalue where the graphs cross. Testing values near x=−2

(−2)^3=−8

(−2)^2−7=−3

Testing values near x=−1.5

(−1.5)^3=−3.375

(−1.5)^2−7=−4.75

5. Refine the solution using a graphing utility or numerical methods. The intersection occurs at approximately x≈−1.663

Final Answer

x≈−1.663

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