Full solution

Q. Solve by completing the square.

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q^2 - 14q - 27 = 0

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

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q =

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Rewrite in standard form: $q^2 - 14q - 27 = 0$ $\newline$ Rewrite the equation in the form of $q^2 + bq = c$ . $\newline$ Add $27$ to both sides to move the constant term to the right side of the equation. $\newline$ $q^2 - 14q - 27 + 27 = 0 + 27$ $\newline$ $q^2 - 14q = 27$
Complete the square: $q^2 - 14q = 27$ $\newline$ Choose the number to add to both sides to complete the square. $\newline$ Since $\left(-\frac{14}{2}\right)^2 = 49$ , add $49$ to both sides. $\newline$ $q^2 - 14q + 49 = 27 + 49$ $\newline$ $q^2 - 14q + 49 = 76$
Factor left side: $q^2 - 14q + 49 = 76$ $\newline$ Identify the equation after factoring the left side. $\newline$ $q^2 - 14q + 49 = 76$ $\newline$ $(q - 7)^2 = 76$
Take square root: $(q − 7)^2 = 76$ $\newline$ Identify the equation after taking the square root on both sides. $\newline$ Take the square root of both sides of the equation. $\newline$ $\sqrt{(q − 7)^2} = \sqrt{76}$ $\newline$ $q − 7 = \pm\sqrt{76}$
Isolate variable: We found: $\newline$ $q - 7 = \pm\sqrt{76}$ $\newline$ Choose the equation after isolating the variable $q$ . $\newline$ To isolate $q$ , add $7$ to both sides of the equation. $\newline$ $q - 7 + 7 = \pm\sqrt{76} + 7$ $\newline$ $q = 7 \pm \sqrt{76}$
Isolate variable: We found: $\newline$ $q - 7 = \pm\sqrt{76}$ $\newline$ Choose the equation after isolating the variable q. $\newline$ To isolate $q$ , add $7$ to both sides of the equation. $\newline$ $q - 7 + 7 = \pm\sqrt{76} + 7$ $\newline$ $q = 7 \pm \sqrt{76}$ We have: $\newline$ $q = 7 \pm \sqrt{76}$ $\newline$ What are the two values of $q$ ? $\newline$ $q = 7 + \sqrt{76}$ implies $q \approx 7 + 8.72$ which is $q \approx 15.72$ . $\newline$ $q$ $0$ implies $q$ $1$ which is $q$ $2$ . $\newline$ Values of $q$ : $q$ $4$ , $q$ $5$