To solve the equation [tex]\( 2\left(x+\frac{3}{4}\right)^2 - 5 = 123 \)[/tex], let's go through the steps in detail:
1. Add 5 to both sides:
[tex]\[
2\left(x+\frac{3}{4}\right)^2 - 5 + 5 = 123 + 5 \implies 2\left(x+\frac{3}{4}\right)^2 = 128
\][/tex]
2. Divide both sides by 2:
[tex]\[
\frac{2\left(x+\frac{3}{4}\right)^2}{2} = \frac{128}{2} \implies \left(x+\frac{3}{4}\right)^2 = 64
\][/tex]
3. Take the square root of both sides:
[tex]\[
x + \frac{3}{4} = \pm \sqrt{64} \implies x + \frac{3}{4} = \pm 8
\][/tex]
4. Subtract [tex]\(\frac{3}{4}\)[/tex] from both sides:
[tex]\[
x = 8 - \frac{3}{4} \quad \text{or} \quad x = -8 - \frac{3}{4}
\][/tex]
Simplifying these expressions:
[tex]\[
x = 8 - \frac{3}{4} = 7.25 \quad \text{and} \quad x = -8 - \frac{3}{4} = -8.75
\][/tex]
So, the solutions to the equation are [tex]\( x = 7.25 \)[/tex] and [tex]\( x = -8.75 \)[/tex].
Therefore, the ordered list of steps that solve the equation is as follows:
1. Add 5 to both sides
2. Divide both sides by 2
3. Take the square root of both sides
4. Subtract [tex]\(\frac{3}{4}\)[/tex] from both sides
