Question

A graph of 2 functions is shown below. graph of function f of x equals negative 11 by 3 multiplied by x plus 11 by 3 and graph of function g of x equals x cubed plus 2 multiplied by x squared minus x minus 2 Which of the following is a solution for f(x) = g(x)? (2 points) x = −2 x = 1 x = 0 x = −1

Answer 1

9514 1404 393

Answer:

(b) x = 1

Explanation:

A graph shows the solution to f(x) = g(x) is x = 1.

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We want to solve ...

g(x) -f(x) = 0

x^3 +2x^2 -x -2 -(-11/3x +11/3) = 0

x^2(x +2) -1(x +2) +11/3(x -1) = 0 . . . . . factor first terms by grouping

(x^2 -1)(x +2) +11/3(x -1) = 0 . . . . . . the difference of squares can be factored

(x -1)(x +1)(x +2) +(x -1)(11/3) = 0 . . . . we see (x-1) is a common factor

(x -1)(x^2 +3x +2 +11/3) = 0

The zero product rule tells us this will be true when x-1 = 0, or x = 1.

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The discriminant of the quadratic factor is ...

b^2 -4ac = 3^2 -4(1)(17/3) = 9 -68/3 = -41/3

This is less than zero, so any other solutions are complex.