Factor 3x^2-12x+8

Problem

3*x^2−12*x+8

Solution

1. Identify the coefficients of the quadratic expression a*x^2+b*x+c Here, a=3 b=−12 and c=8

2. Check for common factors among the terms. The greatest common factor of 3 −12 and 8 is 1 so no terms can be factored out.

3. Calculate the discriminant to determine if the quadratic is factorable over the integers. The discriminant is D=b^2−4*a*c

D=(−12)^2−4*(3)*(8)

D=144−96

D=48

4. Evaluate the result. Since 48 is not a perfect square, the expression cannot be factored into linear binomials with integer or rational coefficients.

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

x=(12±√(,48))/(2*(3))

x=(12±4√(,3))/6

x=2±(2√(,3))/3

6. Construct the factored form using the roots (r_1) and (r_2) in the form a*(x−(r_1))*(x−(r_2))

3*(x−(2+(2√(,3))/3))*(x−(2−(2√(,3))/3))

Final Answer

3*x^2−12*x+8=3*(x−2−(2√(,3))/3)*(x−2+(2√(,3))/3)

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