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$Rewrite the function by completing the square. {:[f(x)=x^(2)-2x+17],[f(x)=(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $f(x) = x^2 - 2x + 17$ , $f(x) = (x + \square)^2 + \square$

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Solve a quadratic equation by completing the square

Full solution

Q. Rewrite the function by completing the square.

\newline

f(x) = x^2 - 2x + 17

,

f(x) = (x + \square)^2 + \square

Identify coefficient of $x$ : We start with the function $f(x) = x^$ $2$ - $2$ x + $17$ and want to rewrite it in the form $f(x) = (x + a)^$ $2$ + b. To complete the square, we need to find a value that, when added and subtracted to the $x^$ $2$ - $2$ x part, creates a perfect square trinomial.
Find value to complete the square: First, we identify the coefficient of $x$ , which is $-2$ . To find the value to complete the square, we take half of this coefficient and square it. This value is $\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1$ .
Add and subtract to create perfect square trinomial: We add and subtract this value inside the function to create a perfect square trinomial. We have to be careful to maintain the equality of the function, so we add $1$ and subtract $1$ inside the function. $\newline$ $f(x) = (x^2 - 2x + 1) + 17 - 1$
Factor and simplify: Now we factor the perfect square trinomial and simplify the constants. $\newline$ $f(x) = (x - 1)^2 + 16$
Rewritten function using completed square: We have successfully rewritten the function by completing the square. The completed square form of the function is $f(x) = (x - 1)^2 + 16$ .

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