Factor x^2+3x+1

Problem

x2+3*x+1

Solution

1. Identify the coefficients of the quadratic expression a*x2+b*x+c where a=1 b=3 and c=1

2. Check for rational roots by calculating the discriminant D=b2−4*a*c

D=3−4*(1)*(1)

D=9−4

D=5

3. Determine factorability over the integers. Since the discriminant 5 is not a perfect square, the expression cannot be factored using integers.

4. Apply the quadratic formula to find the roots x=(−b±√(,D))/(2*a)

x=(−3±√(,5))/(2*(1))

5. Write the factors using the roots (x_1)=(−3+√(,5))/2 and (x_2)=(−3−√(,5))/2 in the form (x−(x_1))*(x−(x_2))

(x−(−3+√(,5))/2)*(x−(−3−√(,5))/2)

6. Simplify the signs within the factors.

(x+(3−√(,5))/2)*(x+(3+√(,5))/2)

Final Answer

x2+3*x+1=(x+(3−√(,5))/2)*(x+(3+√(,5))/2)

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