Using the quadratic formula to solve [tex]\(7x^2 - x = 7\)[/tex], what are the values of [tex]\(x\)[/tex]?

A. [tex]\(\frac{1 \pm \sqrt{195} i}{14}\)[/tex]

B. [tex]\(\frac{1 \pm \sqrt{197}}{14}\)[/tex]

C. [tex]\(\frac{1 \pm \sqrt{195}}{14}\)[/tex]

D. [tex]\(\frac{1 \pm \sqrt{197} i}{14}\)[/tex]



Answer :

To solve the quadratic equation [tex]\(7x^2 - x = 7\)[/tex], we need to follow these steps:

### Step 1: Write the quadratic equation in standard form

First, rewrite the given equation so that all terms are on one side:
[tex]\[ 7x^2 - x - 7 = 0 \][/tex]

Here, the coefficients are:
[tex]\[ a = 7, \quad b = -1, \quad c = -7 \][/tex]

### Step 2: Calculate the discriminant

The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]

Plugging in the coefficients:
[tex]\[ \Delta = (-1)^2 - 4 \cdot 7 \cdot (-7) \][/tex]
[tex]\[ \Delta = 1 + 196 \][/tex]
[tex]\[ \Delta = 197 \][/tex]

### Step 3: Apply the quadratic formula

The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]

Substituting the values:
[tex]\[ x = \frac{-(-1) \pm \sqrt{197}}{2 \cdot 7} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{197}}{14} \][/tex]

### Step 4: Interpret the solutions

The solutions are:
[tex]\[ x_1 = \frac{1 + \sqrt{197}}{14} \][/tex]
[tex]\[ x_2 = \frac{1 - \sqrt{197}}{14} \][/tex]

### Step 5: Compare with the given choices

The given multiple-choice answers are:
1. [tex]\(\frac{1 \pm \sqrt{195} i}{14}\)[/tex]
2. [tex]\(\frac{1 \pm \sqrt{197}}{14}\)[/tex]
3. [tex]\(\frac{1 \pm \sqrt{195}}{14}\)[/tex]
4. [tex]\(\frac{1 \pm \sqrt{197} i}{14}\)[/tex]

From our calculations, we see that the correct choice, matching our solutions, is:
[tex]\[ \boxed{\frac{1 \pm \sqrt{197}}{14}} \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]