The equation `2x^2 + (a - 10)x + 33/2 = 2a` has real roots. Find least positive value of a.

Answer

To solve the equation \(2x^2 + (a - 10)x + \frac{33}{2} = 2a\) for the least positive value of \(a\) such that the equation has real roots, we will follow these steps:

### Step 1: Rearrange the equation
First, we rearrange the equation into standard quadratic form:
\[ 2x^2 + (a - 10)x + \left(\frac{33}{2} - 2a\right) = 0 \]
Here, \(A = 2\), \(B = a - 10\), and \(C = \frac{33}{2} - 2a\).

### Step 2: Determine the discriminant
For the quadratic equation to have real roots, the discriminant \(D\) must be non-negative:
\[ D = B^2 - 4AC \geq 0 \]
Substituting \(A\), \(B\), and \(C\):
\[ D = (a - 10)^2 - 4 \cdot 2 \cdot \left(\frac{33}{2} - 2a\right) \geq 0 \]

### Step 3: Simplify the discriminant
Calculating the discriminant:
\[ D = (a - 10)^2 - 8\left(\frac{33}{2} - 2a\right) \]
\[ = (a - 10)^2 - 8 \cdot \frac{33}{2} + 16a \]
\[ = (a - 10)^2 - 132 + 16a \]
Now, expand \((a - 10)^2\):
\[ = a^2 - 20a + 100 - 132 + 16a \]
\[ = a^2 - 4a - 32 \]

### Step 4: Set up the inequality
We need:
\[ a^2 - 4a - 32 \geq 0 \]

### Step 5: Factor the quadratic
To solve the inequality, we can factor the quadratic:
\[ a^2 - 4a - 32 = (a - 8)(a + 4) \geq 0 \]

### Step 6: Analyze the factors
We find the critical points where the expression equals zero:
\[ a - 8 = 0 \quad \Rightarrow \quad a = 8 \]
\[ a + 4 = 0 \quad \Rightarrow \quad a = -4 \]

### Step 7: Test intervals
We test the intervals determined by the critical points:
1. \(a < -4\) (choose \(a = -5\)): \((-)(-)\) = positive
2. \(-4 < a < 8\) (choose \(a = 0\)): \((-)(+)\) = negative
3. \(a > 8\) (choose \(a = 9\)): \((+)(+)\) = positive

### Step 8: Write the solution set
The solution to the inequality is:
\[ a \in (-\infty, -4] \cup [8, \infty) \]

### Step 9: Find the least positive value of \(a\)
The least positive value of \(a\) from the solution set is:
\[ \boxed{8} \]