Factor x^2-x+6

Problem

x2−x+6

Solution

1. Identify the coefficients of the quadratic expression in the form a*x2+b*x+c where a=1 b=−1 and c=6

2. Determine if there are two integers that multiply to a*c=6 and add to b=−1

3. List the factor pairs of 6 (1,6) (−1,−6) (2,3) and (−2,−3)

4. Check the sums of these pairs: 1+6=7 −1−6=−7 2+3=5 and −2−3=−5

5. Conclude that since no pair of integers sums to −1 the quadratic cannot be factored over the set of integers.

6. Verify using the discriminant D=b2−4*a*c

D=(−1)2−4*(1)*(6)

D=1−24

D=−23

7. Observe that because the discriminant is negative and not a perfect square, the expression is irreducible over the real numbers.

Final Answer

x2−x+6=irreducible

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