Factor x^2+3x+9

Problem

x2+3*x+9

Solution

1. Identify the coefficients of the quadratic expression a*x2+b*x+c where a=1 b=3 and c=9

2. Check for two numbers that multiply to a*c=9 and add to b=3 The pairs of factors for 9 are (1,9) and (3,3) neither of which sum to 3

3. Evaluate the discriminant D=b2−4*a*c to determine if the polynomial can be factored over the set of real numbers.

D=3−4*(1)*(9)

D=9−36

D=−27

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. It is considered an irreducible quadratic over the real numbers.

Final Answer

x2+3*x+9=is irreducible over the real numbers

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