Answers:
(2, -3) and (3, -3)
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Work Shown:
Since f(x) = -3, we'll replace f(x) with -3 and solve for x
f(x) = x^2 - 5x + 3
-3 = x^2 - 5x + 3 .... f(x) replaced with -3
0 = x^2 - 5x + 3 + 3 ... adding 3 to both sides
0 = x^2 - 5x + 6
x^2 - 5x + 6 = 0
(x - 3)(x - 2) = 0 .... factor; you can also apply the quadratic formula
x-3=0 or x-2=0
x = 3 or x = 2
So that explains where that '2' comes from when they listed (2, ?)
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To find what replaces the question mark, we will plug x = 2 into the function
f(x) = x^2 - 5x + 3
f(2) = 2^2 - 5(2) + 3
f(2) = 4 - 10 + 3
f(2) = -3
We could have skipped this portion and said that the solution is (2,-3) but plugging x = 2 into the function to get the output y = -3 helps confirm that x = 2 is one of the x solutions.
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Let's do the same for x = 3
f(x) = x^2 - 5x + 3
f(3) = 3^2 - 5(3) + 3
f(3) = 9 - 15 + 3
f(3) = -3
We end up with the same output, so x = 3 is confirmed as well.
The graph below visually helps reinforce these confirmations. Each solution is an intersection between the two curves.
