Solve by completing the square. $\newline$ $g^2 + 6g - 23 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $g =$ _ or $g =$ _

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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

g^2 + 6g - 23 = 0

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

\newline

g =

_____ or

g =

_____

Rewrite and Add Constant: $g^2 + 6g - 23 = 0$ $\newline$ Rewrite the equation in the form of $x^2 + bx = c$ . $\newline$ Add $23$ to both sides to set the equation up for completing the square. $\newline$ $g^2 + 6g - 23 + 23 = 0 + 23$ $\newline$ $g^2 + 6g = 23$
Complete the Square: $g^2 + 6g = 23$ $\newline$ Choose the number to add to both sides to complete the square. $\newline$ Since $(6/2)^2 = 9$ , add $9$ to both sides. $\newline$ $g^2 + 6g + 9 = 23 + 9$ $\newline$ $g^2 + 6g + 9 = 32$
Factor and Identify: $g^2 + 6g + 9 = 32$ $\newline$ Identify the equation after factoring the left side. $\newline$ $g^2 + 6g + 9 = 32$ $\newline$ $(g + 3)^2 = 32$
Take Square Root: $g + 3)^2 = 32(\newline$ Identify the equation after taking the square root on both sides. $\newline$ Take the square root of both sides of the equation. $\newline$ \$\sqrt{(g + 3)^2} = \sqrt{32}\)$\newline$$g + 3 = \pm\sqrt{32}$$\newline$$g + 3 \approx \pm5.66$ (rounded to the nearest hundredth)
Isolate Variable: We found:$\newline$$g + 3 \approx \pm5.66$$\newline$Choose the equation after isolating the variable $g$.$\newline$To isolate $g$, subtract $3$ from both sides of the equation.$\newline$$g + 3 - 3 \approx \pm5.66 - 3$$\newline$$g \approx -3 \pm5.66$
Isolate Variable: We found:$\newline$$g + 3 \approx \pm5.66$$\newline$Choose the equation after isolating the variable $g$.$\newline$To isolate $g$, subtract $3$ from both sides of the equation.$\newline$$g + 3 - 3 \approx \pm5.66 - 3$$\newline$$g \approx -3 \pm5.66$We have:$\newline$$g \approx -3 \pm5.66$$\newline$What are the two values of $g$?$\newline$$g \approx -3 + 5.66$ implies $g \approx 2.66$.$\newline$$g$$0$ implies $g$$1$.$\newline$Values of $g$: $g$$3$, $g$$4$