Answer:
(-1, 0) and (4, 5)
Explanation:
You want the solution to the simultaneous equations ...
- f(x) = x² -2x -3
- f(x) = x +1
Solution
The function f(x) is equal to itself, so we can write ...
x² -2x -3 = x +1
x² -3x -4 = 0 . . . . . . . . subtract (x+1)
(x -4)(x +1) = 0 . . . . . . . factor
x = 4 or x = -1 . . . . . . . values that make the factors zero
f(x) = x+1 = 5 or 0
The solutions are (x, f(x)) = (-1, 0) and (4, 5).
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Additional comment
There are numerous ways to solve the equations. We like a graphing calculator for its speed and simplicity. The quadratic can be solved using the quadratic formula, completing the square, factoring, graphing, using a solver app or your calculator.
The constants in the binomial factors are factors of -4 that total -3.
-4 = (-4)(1) = (-2)(2) . . . . . . sums of these factors are -3, 0
The factor pair of interest is -4 and 1, giving us the binomial factors ...
(x-4)(x+1) = x² -3x -4.
The "zero product rule" tells you this product is zero only when one of the factors is zero. (x-4) = 0 means x=4, for example.
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