9514 1404 393
Answer:
Step-by-step explanation:
Finding the x-intercepts means finding the solutions to ...
equation = 0
There are a number of ways you can do this, including ...
Often, these can be solved fairly easily by factoring. The methods used for a leading coefficient of 1 and for leading coefficients other than 1 are not so different. In general, you are looking for factors of a number that have some particular sum. Familiarity with multiplication tables is helpful for this.
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1. x^2 -7x -18 = 0
You want factors of -18 that have a sum of -7.
-18 = -18(1) = -9(2) = -6(3) . . . . . sums are -17, -7, -3
The factors -9 and 2 are the ones of interest here. These make up the constants in the binomial factors of the expression:
x^2 -7x -18 = (x -9)(x +2)
The x-intercepts are the values of x that make these factors zero. Basically, they are the opposites of the factor constants:
x -9 = 0 ⇒ x = 9
x +2 = 0 ⇒ x = -2
The x-intercepts are -2 and 9.
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2. 2b^2 +17b +21 = 0
You work this almost the same way, but you're looking for factors of (2)(21) = 42 that have a sum of 17. That is, the leading coefficient multiplies the constant to give the number being factored.
42 = 1(42) = 2(21) = 3(14) = 6(7) . . . . sums are 43, 23, 17, 13
The factors 3 and 14 have a sum of 17.
When the leading coefficient is not 1, several methods are taught for finding the binomial factors. One that works is to rewrite the middle (linear) term using the factors found, then factor by grouping pairs of terms:
(2b^2 +3b) +(14b +21) = 0
b(2b +3) +7(2b +3) = 0
(b +7)(2b +3) = 0
The same deal applies for finding the x-intercepts. They are the values of x that make these factors zero:
b +7 = 0 ⇒ b = -7
2b +3 = 0 ⇒ b = -3/2
The x-intercepts are -1.5 and -7.
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I attached the graph of these expressions to show the correctness of the above, and to illustrate my favorite way to answer these questions: let a graphing calculator show the answer.
