Explain How to Solve 5x² + 3x – 25 by Completing the Square: What Are the Solutions?

To solve the equation 5x² + 3x – 25 = 0 by completing the square, follow these steps:

1. Divide the entire equation by 5:
   This simplifies the equation to: x² + (3/5)x – 5 = 0
2. Rearrange the equation:
   Move the constant term to the other side:
   x² + (3/5)x = 5
3. Complete the square:
   To do this, take half of the coefficient of x (which is 3/5), square it, and add this value to both sides of the equation.
   Half of (3/5) is (3/10) and squaring it gives (9/100).
   Now add (9/100) to both sides:
   x² + (3/5)x + (9/100) = 5 + (9/100)
4. Simplify the equation:
   The left side can be factored as:
   (x + 3/10)² = 500/100 + 9/100 = 509/100
5. Take the square root of both sides:
   x + 3/10 = ±√(509/100)
6. Isolate x:
   x = -3/10 ± √(509)/10

Thus, the solutions to the equation are:

• x = (-3 + √509) / 10
• x = (-3 – √509) / 10