Factor x^2-4x+5

Problem

x2−4*x+5

Solution

1. Identify the coefficients of the quadratic expression in the form a*x2+b*x+c where a=1 b=−4 and c=5

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=(−4)2−4*(1)*(5)=16−20=−4

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 the complex roots: x=(4±√(,−4))/2

6. Simplify the complex roots to x=(4±2*i)/2 which results in x=2+i and x=2−i

7. Write the factored form using the complex roots (x−(2+i))*(x−(2−i))

Final Answer

x2−4*x+5=(x−2−i)*(x−2+i)

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