Factor x^2-4x+1

Problem

x2−4*x+1

Solution

1. Identify the quadratic expression as a*x2+b*x+c where a=1 b=−4 and c=1

2. Check for rational roots using the quadratic formula, since there are no two integers that multiply to 1 and add to −4

3. Apply the quadratic formula to find the roots of the equation x2−4*x+1=0

x=(−b±√(,b2−4*a*c))/(2*a)

4. Substitute the values into the formula:

x=(4±√(,(−4)2−4*(1)*(1)))/(2*(1))

5. Simplify the expression under the radical:

x=(4±√(,16−4))/2

6. Simplify the square root:

x=(4±√(,12))/2

7. Reduce the fraction by simplifying √(,12) to 2√(,3)

x=(4±2√(,3))/2

8. Divide each term by 2 to find the roots:

x=2±√(,3)

9. Write the factors using the roots (r_1)=2+√(,3) and (r_2)=2−√(,3) in the form (x−(r_1))*(x−(r_2))

(x−(2+√(,3)))*(x−(2−√(,3)))

10. Distribute the negative signs:

(x−2−√(,3))*(x−2+√(,3))

Final Answer

x2−4*x+1=(x−2−√(,3))*(x−2+√(,3))

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