Solve : `|x^(2)+x|-5=0 ` for real x

Answer

To solve the equation \( |x^2 + x| - 5 = 0 \) for real \( x \), we can follow these steps:

### Step 1: Rewrite the Equation
We start by rewriting the equation:
\[ |x^2 + x| = 5 \]

### Step 2: Consider Two Cases for the Absolute Value
The absolute value equation \( |A| = B \) can be split into two cases:
1. \( A = B \)
2. \( A = -B \)

In our case, we have:
1. \( x^2 + x = 5 \)
2. \( x^2 + x = -5 \)

### Step 3: Solve the First Case
For the first case:
\[ x^2 + x - 5 = 0 \]
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 1, c = -5 \):
\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-5) = 1 + 20 = 21 \]
Thus, the solutions are:
\[ x = \frac{-1 \pm \sqrt{21}}{2} \]

### Step 4: Solve the Second Case
For the second case:
\[ x^2 + x + 5 = 0 \]
Using the quadratic formula again with \( a = 1, b = 1, c = 5 \):
\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 5 = 1 - 20 = -19 \]
Since the discriminant is negative, this case has no real solutions.

### Step 5: Conclusion
The only real solutions come from the first case:
\[ x = \frac{-1 + \sqrt{21}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{21}}{2} \]

### Final Answer
The real solutions to the equation \( |x^2 + x| - 5 = 0 \) are:
\[ x = \frac{-1 + \sqrt{21}}{2}, \quad x = \frac{-1 - \sqrt{21}}{2} \]