Find the roots of the equations.
Q. `(x^(2)+8)/(11)=5x-x^(2)-5`

Answer

To find the roots of the equation \(\frac{x^2 + 8}{11} = 5x - x^2 - 5\), we will follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by 11 to eliminate the fraction:

\[ x^2 + 8 = 11(5x - x^2 - 5) \]

### Step 2: Distribute on the right side
Distributing \(11\) on the right side gives:

\[ x^2 + 8 = 55x - 11x^2 - 55 \]

### Step 3: Rearrange the equation
Move all terms to one side of the equation:

\[ x^2 + 11x^2 - 55x + 8 + 55 = 0 \]

Combine like terms:

\[ 12x^2 - 55x + 63 = 0 \]

### Step 4: Identify coefficients
In the quadratic equation \(12x^2 - 55x + 63 = 0\), identify the coefficients:
- \(a = 12\)
- \(b = -55\)
- \(c = 63\)

### Step 5: Use the quadratic formula
The roots of a quadratic equation can be found using the formula:

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

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

\[ x = \frac{-(-55) \pm \sqrt{(-55)^2 - 4 \cdot 12 \cdot 63}}{2 \cdot 12} \]

### Step 6: Calculate the discriminant
Calculate \(b^2 - 4ac\):

\[ (-55)^2 = 3025 \]
\[ 4 \cdot 12 \cdot 63 = 3024 \]
\[ b^2 - 4ac = 3025 - 3024 = 1 \]

### Step 7: Substitute back into the formula
Now substitute back into the quadratic formula:

\[ x = \frac{55 \pm \sqrt{1}}{24} \]

### Step 8: Simplify the roots
This gives us two possible values for \(x\):

\[ x = \frac{55 + 1}{24} = \frac{56}{24} = \frac{7}{3} \]
\[ x = \frac{55 - 1}{24} = \frac{54}{24} = \frac{9}{4} \]

### Final Answer
The roots of the equation are:

\[ x = \frac{7}{3} \quad \text{or} \quad x = \frac{9}{4} \]

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