If `y^(2)-2y-3=0`, find the values of y.

Answer

To solve the quadratic equation \( y^2 - 2y - 3 = 0 \), we will factorize the equation and find the values of \( y \).

### Step 1: Write down the equation
We start with the equation:
\[ y^2 - 2y - 3 = 0 \]

### Step 2: Factor the quadratic
To factor the quadratic, we need to find two numbers that multiply to \(-3\) (the constant term) and add up to \(-2\) (the coefficient of \(y\)). The numbers that satisfy these conditions are \(-3\) and \(1\).

So, we can rewrite the equation as:
\[ y^2 - 3y + y - 3 = 0 \]

### Step 3: Group the terms
Next, we group the terms:
\[ (y^2 - 3y) + (y - 3) = 0 \]

### Step 4: Factor by grouping
Now, we can factor out the common terms from each group:
\[ y(y - 3) + 1(y - 3) = 0 \]

### Step 5: Factor out the common binomial
We can factor out the common binomial \((y - 3)\):
\[ (y - 3)(y + 1) = 0 \]

### Step 6: Set each factor to zero
Now, we set each factor equal to zero:
1. \(y - 3 = 0\)
2. \(y + 1 = 0\)

### Step 7: Solve for \(y\)
Solving these equations gives us:
1. \(y = 3\)
2. \(y = -1\)

### Final Answer
Thus, the values of \(y\) are:
\[ y = 3 \quad \text{or} \quad y = -1 \]

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