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]
### 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]
