If the roots, ` x_(1) and x_(2)`, of the quadratic equation ` x^(2) - 2x + c = 0` also satisfy the equation ` 7x_(2) - 4x_(1) = 47`, then which of the following is true ?

Options

A. ` c =- 15`
B. ` x_(1) =- 5, x_(2) = 3`
C. ` x_(1) = 4.5, x_(2) =- 2.5`
D. c = 15

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Identify the given quadratic equation and its roots
The quadratic equation is given as:
\[ x^2 - 2x + c = 0 \]
Let the roots of this equation be \( x_1 \) and \( x_2 \).

### Step 2: Use the relationships for the sum and product of the roots
From Vieta's formulas, we know:
- The sum of the roots \( x_1 + x_2 = -\frac{b}{a} \)
- The product of the roots \( x_1 \cdot x_2 = \frac{c}{a} \)

For our equation:
- \( a = 1 \)
- \( b = -2 \)

Thus, we can calculate:
\[ x_1 + x_2 = -\frac{-2}{1} = 2 \]
This gives us our first equation:
\[ x_1 + x_2 = 2 \quad \text{(Equation 1)} \]

### Step 3: Use the second equation provided
We are also given the equation:
\[ 7x_2 - 4x_1 = 47 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
From Equation 1, we can express \( x_2 \) in terms of \( x_1 \):
\[ x_2 = 2 - x_1 \]

Now, substitute \( x_2 \) into Equation 2:
\[ 7(2 - x_1) - 4x_1 = 47 \]

Expanding this:
\[ 14 - 7x_1 - 4x_1 = 47 \]
Combine like terms:
\[ 14 - 11x_1 = 47 \]

### Step 5: Isolate \( x_1 \)
Now, subtract 14 from both sides:
\[ -11x_1 = 47 - 14 \]
\[ -11x_1 = 33 \]

Divide by -11:
\[ x_1 = -3 \]

### Step 6: Find \( x_2 \)
Now that we have \( x_1 \), we can find \( x_2 \):
\[ x_2 = 2 - x_1 = 2 - (-3) = 2 + 3 = 5 \]

### Step 7: Calculate the value of \( c \)
Using the product of the roots:
\[ x_1 \cdot x_2 = c \]
Substituting the values we found:
\[ (-3) \cdot 5 = c \]
\[ c = -15 \]

### Conclusion
The value of \( c \) is \( -15 \). Therefore, the correct option is that \( c = -15 \).

---