Factor x^2-3x+4

Problem

x2−3*x+4

Solution

1. Identify the coefficients of the quadratic expression in the form a*x2+b*x+c where a=1 b=−3 and c=4

2. Determine if there are two integers that multiply to a*c=4 and add to b=−3

3. List the pairs of factors for 4 (1,4) (−1,−4) (2,2) and (−2,−2)

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

5. Conclude that since no pair of integers sums to −3 the quadratic cannot be factored over the integers.

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

D=(−3)2−4*(1)*(4)

D=9−16

D=−7

7. Observe that because the discriminant is negative, the expression has no real roots and is irreducible over the real numbers.

Final Answer

x2−3*x+4* is irreducible over the integers

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