2

2x2

+2x−0.5=42

We start with the equation:

  22x² + 2x − 0.5 = 42

Step 1. Rearrange the Equation
Subtract 42 from both sides to set the equation equal to 0:
  22x² + 2x − 0.5 − 42 = 0
  22x² + 2x − 42.5 = 0

This is a quadratic equation of the form ax² + bx + c = 0 where:
  a = 22, b = 2, c = −42.5

Step 2. Apply the Quadratic Formula
The quadratic formula is:
  x = [−b ± √(b² − 4ac)] / (2a)

Plug in the values:
  x = [−2 ± √(2² − 4·22·(−42.5))] / (2·22)

Step 3. Evaluate the Discriminant
Compute b² − 4ac:
  b² = 4
  4ac = 4 · 22 · (−42.5) = −3740
Therefore,
  Discriminant = 4 − (−3740) = 4 + 3740 = 3744

Step 4. Simplify the Square Root
We need to simplify √3744. Notice that:
  3744 = 16 × 234
And 234 can be factored as:
  234 = 9 × 26
Thus:
  √3744 = √(16 · 9 · 26) = √16 · √9 · √26 = 4 · 3 · √26 = 12√26

Step 5. Write the Final Expression for x
Now substitute back into the quadratic formula:
  x = [−2 ± 12√26] / 44

We can simplify the fraction by dividing the numerator and the denominator by 2:
  x = [−1 ± 6√26] / 22

So the solutions to the equation are:
  x = (−1 + 6√26) / 22  and  x = (−1 − 6√26) / 22

These are the two solutions for x.