Find the greatest value of 3+5x-`2x^(2)` for all real values of x.

Answer

To find the greatest value of the expression \(3 + 5x - 2x^2\) for all real values of \(x\), we can follow these steps:

### Step 1: Rewrite the expression
Let \(y = 3 + 5x - 2x^2\).

### Step 2: Identify the coefficients
We can rewrite the expression in the standard quadratic form:
\[ y = -2x^2 + 5x + 3 \]
Here, we identify:
- \(a = -2\)
- \(b = 5\)
- \(c = 3\)

### Step 3: Find the vertex of the quadratic
The vertex of a quadratic equation \(ax^2 + bx + c\) occurs at \(x = -\frac{b}{2a}\).
Substituting the values of \(a\) and \(b\):
\[ x = -\frac{5}{2 \times -2} = \frac{5}{4} \]

### Step 4: Substitute \(x\) back into the equation to find \(y\)
Now we substitute \(x = \frac{5}{4}\) back into the equation for \(y\):
\[ y = 3 + 5\left(\frac{5}{4}\right) - 2\left(\frac{5}{4}\right)^2 \]
Calculating each term:
1. \(5 \times \frac{5}{4} = \frac{25}{4}\)
2. \(\left(\frac{5}{4}\right)^2 = \frac{25}{16}\) and \(2 \times \frac{25}{16} = \frac{50}{16} = \frac{25}{8}\)

Now substituting these values into \(y\):
\[ y = 3 + \frac{25}{4} - \frac{25}{8} \]
To combine these, convert \(3\) to a fraction with a denominator of \(8\):
\[ 3 = \frac{24}{8} \]
Now we can combine:
\[ y = \frac{24}{8} + \frac{50}{8} - \frac{25}{8} = \frac{24 + 50 - 25}{8} = \frac{49}{8} \]

### Step 5: Conclusion
Thus, the greatest value of \(3 + 5x - 2x^2\) for all real values of \(x\) is:
\[ \boxed{\frac{49}{8}} \]