Solve for x x^3-5x^2+17x-13=0

Problem

x3−5*x2+17*x−13=0

Solution

1. Identify a possible rational root using the Rational Root Theorem. Testing x=1

1−5*(1)2+17*(1)−13=1−5+17−13=0

Since x=1 is a root, (x−1) is a factor.

2. Divide the polynomial x3−5*x2+17*x−13 by (x−1) using synthetic division or long division.

(x3−5*x2+17*x−13)/(x−1)=x2−4*x+13

3. Factor the resulting quadratic equation x2−4*x+13=0 using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

x=(−(−4)±√(,(−4)2−4*(1)*(13)))/(2*(1))

4. Simplify the expression under the square root.

x=(4±√(,16−52))/2

x=(4±√(,−36))/2

5. Solve for the complex roots using the imaginary unit i=√(,−1)

x=(4±6*i)/2

x=2±3*i

Final Answer

x=1,2+3*i,2−3*i

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