Let's break down the steps to fill in the missing Step 2 and understand the solution for the quadratic equation:
1. Starting with the Given Equation:
[tex]\[
2(x-3)^2 + 6 = 14
\][/tex]
2. Isolate the Squared Term:
To isolate the squared term, we first need to move the constant on the left-hand side to the right-hand side by subtracting 6 from both sides:
[tex]\[
2(x-3)^2 + 6 - 6 = 14 - 6
\][/tex]
This simplifies to:
[tex]\[
2(x-3)^2 = 8
\][/tex]
This is Step 1.
3. Divide Both Sides by 2:
Next, to isolate [tex]\((x-3)^2\)[/tex], we divide both sides of the equation by 2:
[tex]\[
\frac{2(x-3)^2}{2} = \frac{8}{2}
\][/tex]
Simplifying this gives:
[tex]\[
(x-3)^2 = 4
\][/tex]
This would be Step 2.
4. Take the Square Root of Both Sides:
To solve for [tex]\(x-3\)[/tex], we need to take the square root of both sides. Remember that taking the square root of a number will give two results: a positive and a negative root:
[tex]\[
x - 3 = \pm 2
\][/tex]
This simplifies to two equations:
[tex]\[
x - 3 = 2 \quad \text{or} \quad x - 3 = -2
\][/tex]
This is Step 3.
5. Solve for [tex]\(x\)[/tex]:
Now, solve each equation separately:
[tex]\[
x - 3 = 2 \implies x = 2 + 3 \implies x = 5
\][/tex]
[tex]\[
x - 3 = -2 \implies x = -2 + 3 \implies x = 1
\][/tex]
Therefore, the solutions are:
[tex]\[
x = 1 \quad \text{or} \quad x = 5
\][/tex]
This is Step 4.
So, summarizing the steps, Tara's work would look like:
[tex]\[
\begin{array}{ll}
2(x-3)^2+6=14 & \\
2(x-3)^2=8 & \text {Step 1} \\
(x-3)^2=4 & \text {Step 2} \\
x-3= \pm 2 & \text {Step 3} \\
x=1 \text { or } x=5 & \text {Step 4}
\end{array}
\][/tex]
Therefore, the missing Step 2 should be:
[tex]\[
(x-3)^2 = 4
\][/tex]
