Solve by completing the square. $\newline$ $k^2 + 20k = 23$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $k =$ _ or $k =$ _

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Algebra 2

Solve a quadratic equation by completing the square

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Q. Solve by completing the square.

\newline

k^2 + 20k = 23

\newline

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

\newline

k =

_____ or

k =

_____

Write Equation Form: Write the equation in the form of $k^2 + bk = c$ . The given equation is already in this form: $k^2 + 20k = 23$ .
Move Constant Term: Move the constant term to the right side of the equation. $\newline$ Subtract $23$ from both sides to isolate the $k$ terms. $\newline$ $k^2 + 20k - 23 = 0$ $\newline$ $k^2 + 20k = 23$
Complete Square: Complete the square by adding the square of half the coefficient of $k$ to both sides. $\newline$ The coefficient of $k$ is $20$ , so half of it is $10$ . The square of $10$ is $100$ . $\newline$ Add $100$ to both sides of the equation. $\newline$ $k^2 + 20k + 100 = 23 + 100$ $\newline$ $k^2 + 20k + 100 = 123$
Factor Perfect Square: Factor the left side of the equation as a perfect square. The left side is now a perfect square trinomial. $(k + 10)^2 = 123$
Take Square Root: Take the square root of both sides of the equation. $\newline$ $\sqrt{(k + 10)^2} = \pm\sqrt{123}$ $\newline$ $k + 10 = \pm\sqrt{123}$
Solve for k: Solve for k by isolating the variable. $\newline$ Subtract $10$ from both sides of the equation. $\newline$ $k = -10 \pm \sqrt{123}$
Simplify Square Root: Simplify the square root if possible and round to the nearest hundredth if necessary. $\sqrt{123}$ is not a perfect square, so we will use a calculator to approximate the value. $\sqrt{123} \approx 11.09$ $k \approx -10 \pm 11.09$
Find Values of k: Find the two values of $k$ . $k \approx -10 + 11.09$ implies $k \approx 1.09$ . $k \approx -10 - 11.09$ implies $k \approx -21.09$ .Values of $k$ : $1.09$ , $-21.09$