It can be shown that the cubic polynomial equation
will have real roots if the inequality p3/q2 > 27/4 is satisﬁed. Select ﬁve pairs of values for p and q for which this inequality is satisﬁed and hence, using the MATLAB function roots, verify in each case that the roots of the equation are real. In the 16th century the mathematician Ioannes Colla suggested the following problem: Divide 10 into three parts such that they shall be in continued proportion to each other and the product of the ﬁrst two shall be 6. Taking x, y, and z as three parts, this problem can be stated as:
x + y + z =10, x/y=y/z, xy=6
Now by simple manipulation these equations can be expressed in terms of the speciﬁc variable y as:
Clearly if we can solve this equation for y then we can easily ﬁnd the other variables x and z from the original equations. Use the MATLAB function roots to ﬁnd values for y and hence solve Colla’s problem.
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