Question 196269
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Use a substitution.


Let *[tex \LARGE u = x^2], then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^4 - 13x + 36 = u^2 - 13u + 36 = 0]


Factor the quadratic:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (u - 9)(u - 4) = 0]


So, *[tex \LARGE u = 9] or *[tex \LARGE u = 4], but if *[tex \LARGE u = 9] then *[tex \LARGE x^2 = 9] hence *[tex \LARGE x = \pm 3] and if *[tex \LARGE u = 4] then *[tex \LARGE x^2 = 4] hence *[tex \LARGE x = \pm 2]


Therefore the four roots of the quartic are *[tex \LARGE -2,\,2,\,-3,\,\text{and}\,3]


Checking the answer, either by substitution of each of the four values into the original equation or by multiplying:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (x - 2)(x + 2)(x - 3)(x + 3)]


to verify that the completely simplified expansion is identical to the original quartic trinomial is left as an exercise for the student.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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