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Tara solved a quadratic equation. Her work is shown below, with Step 2 missing.

What could Tara have written as the result from Step 2?

[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]

Sagot :

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]