Factor x^2+2x+2

Problem

x2+2*x+2

Solution

1. Identify the coefficients of the quadratic expression a*x2+b*x+c where a=1 b=2 and c=2

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

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

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

5. Apply the quadratic formula x=(−b±√(,D))/(2*a) to find complex roots: x=(−2±√(,−4))/(2*(1))=(−2±2*i)/2=−1±i

6. Write the factored form using the complex roots (r_1)=−1+i and (r_2)=−1−i in the form (x−(r_1))*(x−(r_2))

Final Answer

x2+2*x+2=(x+1−i)*(x+1+i)

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