first equation is
(VIL -VT)^2/2 = [(VDD - VIL - VT) (VDD - VOUT) - (VDD - VOUT)^2/2]
second equation is
VOUT = VIL + VDD/2
now substitute second equation in first equation and give me resultant VIL equation

Let's begin by substituting the second equation, \( V_{\text{OUT}} = V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \), into the first equation and then solving for \( V_{\text{IL}} \).

### Step 1: First Equation
\[
(V_{\text{IL}} - V_{\text{T}})^2 / 2 = \left[ (V_{\text{DD}} - V_{\text{IL}} - V_{\text{T}}) (V_{\text{DD}} - V_{\text{OUT}}) - \frac{(V_{\text{DD}} - V_{\text{OUT}})^2}{2} \right]
\]

### Step 2: Substitute \( V_{\text{OUT}} = V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \)
Now, substitute this into the equation:
\[
(V_{\text{IL}} - V_{\text{T}})^2 / 2 = \left[ (V_{\text{DD}} - V_{\text{IL}} - V_{\text{T}}) \left( V_{\text{DD}} - \left( V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \right) \right) - \frac{\left( V_{\text{DD}} - \left( V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \right) \right)^2}{2} \right]
\]

### Step 3: Simplify \( V_{\text{DD}} - V_{\text{OUT}} \)
Simplifying \( V_{\text{DD}} - V_{\text{OUT}} \) using \( V_{\text{OUT}} = V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \), we get:
\[
V_{\text{DD}} - V_{\text{OUT}} = V_{\text{DD}} - \left( V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \right) = \frac{V_{\text{DD}}}{2} - V_{\text{IL}}
\]

### Step 4: Substitute into the Equation
Now substitute this into the original equation:
\[
(V_{\text{IL}} - V_{\text{T}})^2 / 2 = \left[ (V_{\text{DD}} - V_{\text{IL}} - V_{\text{T}}) \left( \frac{V_{\text{DD}}}{2} - V_{\text{IL}} \right) - \frac{\left( \frac{V_{\text{DD}}}{2} - V_{\text{IL}} \right)^2}{2} \right]
\]

### Step 5: Expand and Simplify
Now expand the terms and simplify the equation to isolate \( V_{\text{IL}} \). After expanding both sides, collect terms of \( V_{\text{IL}} \) and solve.

Let me handle this algebraic expansion to get the resultant equation.The solution for \( V_{\text{IL}} \) after substituting the second equation into the first is:

\[
V_{\text{IL}} = \frac{3V_{\text{DD}}}{8} + \frac{V_{\text{T}}}{4}
\]

Would you like a detailed breakdown of the steps involved in solving this, or do you have any other questions?

Here are 5 related questions:
1. How does the threshold voltage \( V_T \) affect the value of \( V_{\text{IL}} \)?
2. What physical meaning does \( V_{\text{IL}} \) represent in this context?
3. How would the equation change if the source voltage \( V_{\text{DD}} \) were increased?
4. Can you derive the relationship between \( V_{\text{OUT}} \) and \( V_T \) using these equations?
5. How does this result apply to transistor behavior in digital circuits?

**Tip:** Expanding and simplifying equations early on can save a lot of time when solving for variables.

The query submitted by the user: first equation is (VIL - VT)^2/2 = [(VDD - VIL - VT) (VDD - VOUT) - (VDD - VOUT)^2/2] second equation is VOUT = VIL + VDD/2 now substitute second equation in first equation and give me resultant VIL equation.

Learn how to solve for VIL by substituting the equation VOUT = VIL + VDD / 2 into the original equation (VIL - VT)^2/2 = [(VDD - VIL - VT)(VDD - VOUT) - (VDD - VOUT)^2/2]. This detailed guide walks through algebraic substitution and expansion. Suitable for college-level engineering or advanced physics students.

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