Explanation Given information: The equation for the vertices is V = x 2 − 14 x , equation for the edges is E = x 2 − 12 x − 8 and the equation for the regions is F = x 2 − 11 x − 36 . Calculation: The vertices are the corners of the figure where two lines, edges or curves meet with each other. Edges are the lines and curves between two vertices. The regions is the area on the plane that is surrounded and bounded by edges and it is denoted as F . The Euler characteristic theorem is V − E + F = 2 . Now substitute the given equation of the vertices, edges and regions in the Euler characteristic theorem V − E + F = 2 as, ( x 2 − 14 x ) − ( x 2 − 12 x − 8 ) + ( x 2 − 11 x − 36 ) = 2 x 2 − 14 x − x 2 + 12 x + 8 + x 2 − 11 x − 36 = 2 x 2 − 13 x − 28 = 2 x 2 − 13 x − 30 = 0 Now make factors of the resultant quadratic equation by middle splitting term method. x 2 − 13 x − 30 = 0 x 2 − 15 x + 2 x − 30 = 0 x ( x − 15 ) + 2 ( x − 15 ) = 0 ( x − 15 ) ( x + 2 ) = 0 Now by the use of zero factor property the value of z can be found as, ( x − 15 ) ( x + 2 ) = 0 x − 15 = 0 or x + 2 = 0 x = 15 or x = − 2 Therefore, there are two possibilities for x that is either − 2 or 15

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ISBN: 9781119760061

Author: Burger

Publisher: WILEY

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Chapter 6.3, Problem 37MS

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To find: The number of vertices, edges and the regions.

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Chapter 6.3, Problem 36MS

Chapter 6.3, Problem 38MS

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The number of vertices, edges and the regions.

Solution Summary: The author calculates the number of vertices, edges, and regions by using the Euler characteristic theorem.

To find: The number of vertices, edges and the regions.

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