Factor x^2+2x+3

Problem

x2+2*x+3

Solution

1. Identify the coefficients of the quadratic expression in the form a*x2+b*x+c Here, a=1 b=2 and c=3

2. Calculate the discriminant to determine if the quadratic can be factored over the real numbers using the formula D=b2−4*a*c

3. Substitute the values into the discriminant formula: D=2−4*(1)*(3)=4−12=−8

4. Conclude that since the discriminant is negative (D<0, there are no real roots, and the expression is irreducible over the set of real numbers.

5. Apply the quadratic formula x=(−b±√(,D))/(2*a) to find complex factors if required.

6. Solve for the roots: x=(−2±√(,−8))/2=(−2±2*i√(,2))/2=−1±i√(,2)

7. Write the factored form using the roots (r_1)=−1+i√(,2) and (r_2)=−1−i√(,2) in the form (x−(r_1))*(x−(r_2))

Final Answer

x2+2*x+3=(x+1−i√(,2))*(x+1+i√(,2))

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