Factor x^2-2x+3

Problem

x2−2*x+3

Solution

1. Identify the coefficients of the quadratic expression a*x2+b*x+c where 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)2−4*(1)*(3)=4−12=−8

4. Conclude that since the discriminant is negative (D<0, the quadratic has no real roots and cannot be factored into linear factors with real coefficients.

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

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

7. Write the factored form using the complex roots (x−(r_1))*(x−(r_2))

Final Answer

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

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