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$Rewrite the function by completing the square. {:[f(x)=x^(2)+16 x-46],[f(x)=(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $f(x) = x^2 + 16x - 46$ , $f(x) = (x+\square)^2 + \square$

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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 + 16x - 46

,

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

Step $1$ : Rewrite the function: We start with the function $f(x) = x^2 + 16x - 46$ and want to rewrite it in the form $f(x) = (x + a)^2 + b$ . $\newline$ First, we need to complete the square for the $x^2 + 16x$ part of the function. $\newline$ To do this, we take half of the coefficient of $x$ , which is $16$ , and square it. $\newline$ $(\frac{16}{2})^2 = 8^2 = 64$ .
Step $2$ : Complete the square: We then add and subtract this number, $64$ , inside the function to complete the square without changing the function's value. $\newline$ $f(x) = x^2 + 16x + 64 - 64 - 46.$
Step $3$ : Add and subtract to complete the square: Now we can rewrite the function by factoring the perfect square trinomial and combining the constants. $\newline$ $f(x) = (x + 8)^2 - 64 - 46$ .
Step $4$ : Factor the perfect square trinomial: Combine the constants $-64$ and $-46$ to get $-110$ . $\newline$ f(x) = (x + $8$ )^ $2$ - $110$ .

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