Full solution

Q. Solve by completing the square.

\newline

u^2 + 10u - 9 = 0

\newline

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

\newline

u =

_____ or

u =

_____

Rewrite in standard form: $u^2 + 10u - 9 = 0$ $\newline$ Rewrite the equation in the form of $x^2 + bx = c$ . $\newline$ Add $9$ to both sides to move the constant term to the right side of the equation. $\newline$ $u^2 + 10u - 9 + 9 = 0 + 9$ $\newline$ $u^2 + 10u = 9$
Complete the square: $u^2 + 10u = 9$ $\newline$ Choose the equation after completing the square. $\newline$ Since $(10/2)^2 = 25$ , add $25$ to both sides to complete the square. $\newline$ $u^2 + 10u + 25 = 9 + 25$ $\newline$ $u^2 + 10u + 25 = 34$
Factor left side: $u^2 + 10u + 25 = 34$ $\newline$ Identify the equation after factoring the left side. $\newline$ $u^2 + 10u + 25 = 34$ $\newline$ $(u + \frac{10}{2})^2 = 34$ $\newline$ $(u + 5)^2 = 34$
Take square root: $(u + 5)^2 = 34$ $\newline$ Identify the equation after taking the square root on both sides. $\newline$ Take the square root of both sides of the equation. $\newline$ $\sqrt{(u + 5)^2} = \pm\sqrt{34}$ $\newline$ u + $5$ = \pm\sqrt{ $34$ }
Isolate variable: We found: $\newline$ $u + 5 = \pm\sqrt{34}$ $\newline$ Choose the equation after isolating the variable $u$ . $\newline$ To isolate $u$ , subtract $5$ from both sides of the equation. $\newline$ $u + 5 - 5 = \pm\sqrt{34} - 5$ $\newline$ $u = -5 \pm \sqrt{34}$
Isolate variable: We found: $\newline$ $u + 5 = \pm\sqrt{34}$ $\newline$ Choose the equation after isolating the variable $u$ . $\newline$ To isolate $u$ , subtract $5$ from both sides of the equation. $\newline$ $u + 5 - 5 = \pm\sqrt{34} - 5$ $\newline$ $u = -5 \pm \sqrt{34}$ We have: $\newline$ $u = -5 \pm \sqrt{34}$ $\newline$ What are the two values of $u$ ? $\newline$ $u = -5 + \sqrt{34}$ implies $u \approx -5 + 5.83$ which is $u$ $0$ . $\newline$ $u$ $1$ implies $u$ $2$ which is $u$ $3$ . $\newline$ Values of $u$ : $u$ $5$ , $u$ $6$