solve 1/2(x-3)^2-5=3 using quadratic formula

To solve the equation \(\frac{1}{2}(x - 3)^2 - 5 = 3\) using the quadratic formula, follow these steps:

### Step 1: Isolate the quadratic term
First, move the constants to the other side of the equation:
\[
\frac{1}{2}(x - 3)^2 - 5 = 3
\]
Add 5 to both sides:
\[
\frac{1}{2}(x - 3)^2 = 8
\]

### Step 2: Eliminate the fraction
Multiply both sides by 2 to eliminate the fraction:
\[
(x - 3)^2 = 16
\]

### Step 3: Solve for \(x - 3\)
Take the square root of both sides:
\[
x - 3 = \pm 4
\]
So, you have two possible cases:
\[
x - 3 = 4 \quad \text{or} \quad x - 3 = -4
\]

### Step 4: Solve for \(x\)
Now, solve for \(x\) in each case:
1. \(x - 3 = 4 \Rightarrow x = 4 + 3 = 7\)
2. \(x - 3 = -4 \Rightarrow x = -4 + 3 = -1\)

### Solutions:
The solutions are:
\[
x = 7 \quad \text{or} \quad x = -1
\]

Would you like more details or have any questions about this process?

Here are 5 related questions to deepen understanding:
1. How would the solution change if the constant 3 on the right were a different number?
2. How can you verify the solutions to this equation by substituting them back into the original equation?
3. Can you solve this equation graphically? What would the graph look like?
4. What would happen if the coefficient of \((x-3)^2\) were something other than \(\frac{1}{2}\)?
5. How would you rewrite this equation if you wanted to avoid using the quadratic formula?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Learn how to solve the equation (1/2)(x-3)^2 - 5 = 3 using the quadratic formula with a detailed, step-by-step solution. This problem covers key algebraic concepts such as quadratic equations and solving by square roots. Ideal for students in Grades 8-10.

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