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$Rewrite the function by completing the square. {:[f(x)=x^(2)+14 x+8],[f(x)=(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $f(x) = x^2 + 14x + 8$ $\newline$ $f(x) = (x+\square)^2 + \square$

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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 + 14x + 8

\newline

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

Given quadratic function: We start with the given quadratic function: $\newline$ $f(x) = x^2 + 14x + 8$ $\newline$ To complete the square, we need to form a perfect square trinomial from the $x^2$ and $x$ terms.
Completing the square: First, we identify the coefficient of the $x$ term, which is $14$ . We then take half of this coefficient and square it to find the number that we need to add and subtract to complete the square. $\newline$ $\left(\frac{$ $14$ }{ $2$ }\right)^ $2$ = $7$ ^ $2$ = $49$
Adding and subtracting: We add and subtract $49$ inside the function to complete the square: $\newline$ f(x) = x^ $2$ + $14$ x + $49$ - $49$ + $8$
Rewriting the function: Now we can rewrite the function by grouping the perfect square trinomial and combining the constants: $\newline$ f(x) = $(x^2 + 14x + 49) - 49 + 8$ $\newline$ f(x) = $(x + 7)^2 - 41$
Final answer: The function is now rewritten in the completed square form: $\newline$ f(x) = (x + $7$ )^ $2$ - $41$ $\newline$ This is the final answer.

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