Answers:
- b = -19
- b = -11
- b = -9
- b = 19
- b = 11
- b = 9
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Explanation:
Here are all the ways to multiply to 18 when using integers only:
- -1*(-18) = 18
- -2*(-9) = 18
- -3*(-6) = 18
- 1*18 = 18
- 2*9 = 18
- 3*6 = 18
Sum each pair of factors to find out a possible value of b.
- -1 + (-18) = -19
- -2 + (-9) = -11
- -3 + (-6) = -9
- 1 + 18 = 19
- 2 + 9 = 11
- 3 + 6 = 9
Therefore, the possible values of b are
- b = -19
- b = -11
- b = -9
- b = 19
- b = 11
- b = 9
which are the final answers.
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An example:
Let's say b = 11. This would mean [tex]x^2+bx+18[/tex] becomes [tex]x^2+11x+18[/tex]
It would factor to [tex](x+2)(x+9)[/tex] since it was stated earlier that:
2+9 = 11
2 * 9 = 18
You can use the FOIL rule, distributive property, or the box method to confirm that [tex]x^2+11x+18 = (x+2)(x+9)[/tex] is a true equation for all real numbers x.
This same idea applies for the other values of b.
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If you're curious why this works, consider multiplying the two factors (x+p) and (x+q)
Use the FOIL rule to get [tex](x+p)(x+q) = x^2+qx+px+pq = x^2+(p+q)x + pq[/tex]
The middle term [tex](p+q)x[/tex] has the components add to the coefficient, while those same two components multiply to get the last term. This is why when factoring we're looking for two numbers that multiply to 18, and also add to the value of b (which in the case of the last example was 11).
