Solve: `2x^(2)-5x+2=(5-sqrt(9+8x))/4`, where `xlt5/4`.

Explanation

To solve the equation \( 2x^2 - 5x + 2 = \frac{5 - \sqrt{9 + 8x}}{4} \) under the condition \( x < \frac{5}{4} \), we will follow these steps:

### Step 1: Clear the Fraction
Multiply both sides of the equation by 4 to eliminate the fraction:
\[ 4(2x^2 - 5x + 2) = 5 - \sqrt{9 + 8x} \]
This simplifies to:
\[ 8x^2 - 20x + 8 = 5 - \sqrt{9 + 8x} \]

### Step 2: Rearrange the Equation
Rearranging gives:
\[ 8x^2 - 20x + 8 - 5 = -\sqrt{9 + 8x} \]
This simplifies to:
\[ 8x^2 - 20x + 3 = -\sqrt{9 + 8x} \]

### Step 3: Square Both Sides
Square both sides to eliminate the square root (note that squaring can introduce extraneous solutions):
\[ (8x^2 - 20x + 3)^2 = (9 + 8x) \]

### Step 4: Expand Both Sides
Expanding the left side:
\[ (8x^2 - 20x + 3)(8x^2 - 20x + 3) = 64x^4 - 320x^3 + 48x^2 + 120x + 9 \]
The right side is:
\[ 9 + 8x \]

### Step 5: Set the Equation to Zero
Combine all terms:
\[ 64x^4 - 320x^3 + 48x^2 + 120x + 9 - 9 - 8x = 0 \]
This simplifies to:
\[ 64x^4 - 320x^3 + 48x^2 + 112x = 0 \]

### Step 6: Factor Out Common Terms
Factor out \( 16x \):
\[ 16x(4x^3 - 20x^2 + 3x + 7) = 0 \]

### Step 7: Solve for Roots
From \( 16x = 0 \), we find:
\[ x = 0 \]
Now, we need to solve the cubic equation \( 4x^3 - 20x^2 + 3x + 7 = 0 \).

### Step 8: Use the Rational Root Theorem or Synthetic Division
Testing possible rational roots, we find that \( x = 2 \) is a root. We can factor the cubic:
\[ 4x^3 - 20x^2 + 3x + 7 = (x - 2)(4x^2 - 12x - 3) \]

### Step 9: Solve the Quadratic
Now, we solve the quadratic \( 4x^2 - 12x - 3 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot (-3)}}{2 \cdot 4} \]
Calculating the discriminant:
\[ 144 + 48 = 192 \]
So,
\[ x = \frac{12 \pm \sqrt{192}}{8} = \frac{12 \pm 8\sqrt{3}}{8} = \frac{3 \pm 2\sqrt{3}}{2} \]

### Step 10: Check Validity of Solutions
We have three potential solutions: \( x = 0 \), \( x = 2 \), and \( x = \frac{3 \pm 2\sqrt{3}}{2} \). However, we need to check which of these satisfy the condition \( x < \frac{5}{4} \).

- \( x = 0 \) is valid.
- \( x = 2 \) is not valid because \( 2 > \frac{5}{4} \).
- \( x = \frac{3 - 2\sqrt{3}}{2} \) needs checking:
- Calculate \( \sqrt{3} \approx 1.732 \), so \( 3 - 2\sqrt{3} \approx 3 - 3.464 \approx -0.464 \), thus \( \frac{3 - 2\sqrt{3}}{2} < 0 \).
- \( x = \frac{3 + 2\sqrt{3}}{2} \) is also not valid as it will be greater than \( \frac{5}{4} \).

### Conclusion
The only solution that satisfies the condition \( x < \frac{5}{4} \) is:
\[ \boxed{0} \]