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$Rewrite the function by completing the square. {:[f(x)=x^(2)+12 x+7],[f(x)=(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $f(x) = x^2 + 12x + 7$ , $f(x) = (x+\square)^2 + \square$

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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 + 12x + 7

,

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

Given quadratic function: We start with the given quadratic function: $\newline$ f(x) = $x^2 + 12x + 7$ $\newline$ To complete the square, we need to find a number to add and subtract to the function that will allow us to write it in the form of $(x + a)^2 + b$ .
Completing the square: First, we identify the coefficient of $x$ , which is $12$ . To complete the square, we take half of this coefficient and square it. This gives us $(\frac{12}{2})^2 = 6^2 = 36$ .
Identifying the coefficient: We add and subtract this number $36$ inside the function to maintain the equality: $\newline$ $f(x) = x^2 + 12x + 36 - 36 + 7$
Adding and subtracting: Now we can rewrite the function by grouping the perfect square trinomial and the constants: $\newline$ f(x) = $(x^2 + 12x + 36) - 36 + 7$
Rewriting the function: The perfect square trinomial $(x^2 + 12x + 36)$ can be factored into $(x + 6)^2$ : $\newline$ f(x) = (x + $6$ )^ $2$ - $36$ + $7$
Factoring the perfect square trinomial: Finally, we combine the constants $-36$ and $+7$ to simplify the function: $\newline$ $f(x) = (x + 6)^2 - 29$

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