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$Rewrite the function by completing the square. {:[f(x)=x^(2)+8x+4],[f(x)=(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $f(x) = x^2 + 8x + 4$ $\newline$ $f(x) = (x + \Box)^2 + \Box$

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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 + 8x + 4

\newline

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

Given quadratic function: We start with the given quadratic function: $\newline$ f(x) = $x^2 + 8x + 4$ $\newline$ To complete the square, we need to form a perfect square trinomial from the quadratic and linear terms.
Forming a perfect square trinomial: The coefficient of $x$ is $8$ . To form a perfect square trinomial, we take half of the coefficient of $x$ , which is $\frac{8}{2} = 4$ , and then square it to get $4^2 = 16$ .
Adding and subtracting to maintain equality: We add and subtract this number $16$ inside the function to maintain the equality: $\newline$ $f(x) = x^2 + 8x + 16 - 16 + 4$
Rewriting the function by grouping: Now we can rewrite the function by grouping the perfect square trinomial and the constants: $\newline$ $f(x) = (x^2 + 8x + 16) - 16 + 4$
Factoring the perfect square trinomial: The expression in the parentheses is a perfect square trinomial, which can be factored into $(x + 4)^2$ : $\newline$ f(x) = $(x + 4)^2 - 16 + 4$
Combining the constants: Finally, we combine the constants $-16$ and $+4$ to get $-12$ : $\newline$ $f(x) = (x + 4)^2 - 12$

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