`x ^(2) + 7x +1 =2x ^(2)-4x +3`
Which of the following is a value of x that is valid in the above equation ?

Options

A. `5.5 - sqrt(28.25)`
B. `sqrt(5.5)`
C. `sqrt(30.25)`
D. `5.5+sqrt(30.25)`

Explanation

To solve the equation \( x^2 + 7x + 1 = 2x^2 - 4x + 3 \), we will follow these steps:

### Step 1: Rearrange the equation
We will move all terms to one side of the equation to set it to zero.

\[ x^2 + 7x + 1 - 2x^2 + 4x - 3 = 0 \]

### Step 2: Combine like terms
Now, we will combine the like terms:

\[ -x^2 + 11x - 2 = 0 \]

Multiplying through by -1 to make the leading coefficient positive:

\[ x^2 - 11x + 2 = 0 \]

### Step 3: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify:

- \( a = 1 \)
- \( b = -11 \)
- \( c = 2 \)

### Step 4: Use the quadratic formula
The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]

### Step 5: Calculate the discriminant
Calculating the discriminant:

\[ b^2 - 4ac = 121 - 8 = 113 \]

### Step 6: Substitute back into the formula
Now substituting back into the quadratic formula:

\[ x = \frac{11 \pm \sqrt{113}}{2} \]

### Step 7: Simplify the expression
This gives us two potential solutions:

\[ x = \frac{11 + \sqrt{113}}{2} \quad \text{and} \quad x = \frac{11 - \sqrt{113}}{2} \]

### Step 8: Approximate the values
Calculating the approximate values:

1. \( \sqrt{113} \approx 10.630 \)
2. Therefore, \( x \approx \frac{11 + 10.630}{2} \approx 10.815 \)
3. And \( x \approx \frac{11 - 10.630}{2} \approx 0.185 \)

### Conclusion
Thus, the values of \( x \) that satisfy the equation are approximately \( 10.815 \) and \( 0.185 \).