Answer :
We express the quadratic functions f(x) = 3x² - 30x + 82 and f(x) = 3x² + 24x + 3 in standard form by completing the square to get f(x) = 3(x - 5)² + 7 and f(x) = 3(x + 4)² - 45 respectively.
To express a quadratic function in standard form, you will need to complete the square. The standard form of a quadratic function is f(x) = an (x - h)² + k where a is the coefficient of x², 'h' is the value that completes the square in the parenthesis, and 'k' is the y-coordinate of the vertex.
a. Let's take the first function f(x) = 3x² - 30x + 82.
First, it is necessary to factor out the 3: f(x) = 3(x² - 10x) + 82.
The next step is to complete the square within the parenthesis.
We take half of -10 which is -5, square it to get 25, and add and subtract it inside the parenthesis: f(x) = 3[(x² - 10x + 25) - 25] + 82. This simplifies to f(x) = 3(x - 5)² + 7.
b. Using the same method for the second function f(x) = 3x² + 24x + 3, we first factor out the 3: f(x) = 3(x² + 8x) + 3.
Then we complete the square: f(x) = 3[(x² + 8x+ 16) - 16] + 3 = 3(x + 4)² - 45.
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