Solve (using formula) the equation : `x/(x+1)+(x+1)/x=2(4)/15`

Answer

To solve the equation \( \frac{x}{x+1} + \frac{x+1}{x} = \frac{2 \cdot 4}{15} \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation clearly:
\[ \frac{x}{x+1} + \frac{x+1}{x} = \frac{8}{15} \]

### Step 2: Find a common denominator
The left-hand side of the equation consists of two fractions. To combine them, we need a common denominator, which is \( x(x+1) \):
\[ \frac{x^2 + (x+1)^2}{x(x+1)} = \frac{8}{15} \]

### Step 3: Expand the numerator
Now, expand \( (x+1)^2 \):
\[ (x+1)^2 = x^2 + 2x + 1 \]
Thus, the numerator becomes:
\[ x^2 + (x^2 + 2x + 1) = 2x^2 + 2x + 1 \]
So, we have:
\[ \frac{2x^2 + 2x + 1}{x(x+1)} = \frac{8}{15} \]

### Step 4: Cross-multiply
Cross-multiply to eliminate the fractions:
\[ 15(2x^2 + 2x + 1) = 8x(x + 1) \]

### Step 5: Expand both sides
Expand both sides:
\[ 30x^2 + 30x + 15 = 8x^2 + 8x \]

### Step 6: Rearrange the equation
Move all terms to one side:
\[ 30x^2 + 30x + 15 - 8x^2 - 8x = 0 \]
This simplifies to:
\[ 22x^2 + 22x + 15 = 0 \]

### Step 7: Use the quadratic formula
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 22 \), \( b = 22 \), and \( c = 15 \).

### Step 8: Calculate the discriminant
Calculate \( b^2 - 4ac \):
\[ b^2 - 4ac = 22^2 - 4 \cdot 22 \cdot 15 = 484 - 1320 = -836 \]

### Step 9: Solve for x
Since the discriminant is negative, the equation has no real solutions. Thus, we conclude:
\[ \text{No real solutions for } x. \]