An equation equivalent to the quadratic equation `x^(2)-6x+5=0` is

Options

A. `6x^(2)-5x+1=0`
B. `x^(2)-5x+6=0`
C. `5x^(2)-6x+1=0`
D. `|x-3|=2`

Answer

To find an equation equivalent to the quadratic equation \(x^2 - 6x + 5 = 0\), we will first determine the roots of the given equation and then find an equivalent equation based on those roots.

### Step 1: Identify the quadratic equation
The given quadratic equation is:
\[ x^2 - 6x + 5 = 0 \]

### Step 2: Factor the quadratic equation
To factor the equation, we need to find two numbers that multiply to \(5\) (the constant term) and add up to \(-6\) (the coefficient of \(x\)).

The factors of \(5\) that meet these criteria are \(-5\) and \(-1\), since:
\[ -5 \times -1 = 5 \quad \text{and} \quad -5 + (-1) = -6 \]

Thus, we can factor the equation as:
\[ (x - 5)(x - 1) = 0 \]

### Step 3: Set each factor to zero to find the roots
Setting each factor equal to zero gives us the roots:
1. \(x - 5 = 0 \Rightarrow x = 5\)
2. \(x - 1 = 0 \Rightarrow x = 1\)

So, the roots of the equation are \(x = 1\) and \(x = 5\).

### Step 4: Formulate equivalent equations
An equivalent equation will have the same roots. We can create an equivalent equation using the roots we found. One common way to express this is to use the factored form:
\[ (x - 1)(x - 5) = 0 \]

### Step 5: Expand the factored form
Expanding the factored form gives:
\[ x^2 - 6x + 5 = 0 \]
This is the same as our original equation, confirming that it is indeed equivalent.

### Step 6: Check for other equivalent forms
We can also express the equivalent equation in different forms. For example, we can manipulate the equation:
\[ |x - 3| = 2 \]
This equation also has the roots \(x = 1\) and \(x = 5\) because:
- For \(x = 1\): \(|1 - 3| = 2\)
- For \(x = 5\): \(|5 - 3| = 2\)

### Conclusion
Thus, an equivalent equation to the quadratic equation \(x^2 - 6x + 5 = 0\) is:
\[ |x - 3| = 2 \]