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$Rewrite the function by completing the square. {:[f(x)=x^(2)+20 x-86],[f(x)=(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $f(x) = x^2 + 20x - 86$ $\newline$ $f(x) = (x + \square)^2 + \square$

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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 + 20x - 86

\newline

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

Given quadratic function: We start with the given quadratic function: $\newline$ f(x) = x^ $2$ + $20$ x - $86$ $\newline$ To complete the square, we need to form a perfect square trinomial from the x^ $2$ and x terms. We will find the value to add and subtract to complete the square.
Completing the square: The coefficient of x is $20$ . To complete the square, we take half of the coefficient of x, square it, and add it to and subtract it from the equation. This value is $(\frac{20}{2})^2 = 10^2 = 100$ .
Adding and subtracting $100$ : We add and subtract $100$ inside the function: $\newline$ f(x) = $(x^2 + 20x + 100) - 100 - 86$ $\newline$ Now we have a perfect square trinomial $x^2 + 20x + 100$ , which can be factored into $(x + 10)^2$ .
Factoring the perfect square trinomial: We simplify the constants $-100$ and $-86$ : $\newline$ f(x) = (x + $10$ )^ $2$ - $100$ - $86$ $\newline$ f(x) = (x + $10$ )^ $2$ - $186$ $\newline$ Now the function is written in the completed square form.

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