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We are asked to find the sum of constants \(a + b\) for a line \(y = ax + b\) that is tangent to two curves \(y = x^3\) and \(y = 3x^2 - 4\) at the same point \(x = t\).

### Step 1: Condition for Tangency
For the line \(y = ax + b\) to be tangent to both curves at \(x = t\), it must satisfy two conditions:
1. **The point of tangency**: The line should pass through the points on both curves at \(x = t\), meaning the \(y\)-values should match.
   - \( t^3 = at + b \) (from \(y = x^3\))
   - \( 3t^2 - 4 = at + b \) (from \(y = 3x^2 - 4\))

2. **Matching the slopes**: The derivative of the curves at \(x = t\) should equal the slope of the line, which is \(a\).
   - Slope of \(y = x^3\) at \(x = t\): \(3t^2\)
   - Slope of \(y = 3x^2 - 4\) at \(x = t\): \(6t\)

Thus, the following system of equations emerges:
\[
3t^2 = a
\]
\[
6t = a
\]
From this, we see \(a = 3t^2 = 6t\).

### Step 2: Solve for \(t\)
Equating the expressions for \(a\):
\[
3t^2 = 6t
\]
Dividing through by \(t\) (assuming \(t \neq 0\)):
\[
3t = 6 \quad \Rightarrow \quad t = 2
\]

### Step 3: Find \(a\) and \(b\)
Now that we know \(t = 2\), we can substitute it back into the expressions for \(a\) and solve for \(b\):
\[
a = 3t^2 = 3(2)^2 = 12
\]

Next, substitute \(t = 2\) into either of the original point equations (say, \(t^3 = at + b\)) to find \(b\):
\[
2^3 = 12(2) + b
\]
\[
8 = 24 + b \quad \Rightarrow \quad b = -16
\]

### Step 4: Calculate \(a + b\)
\[
a + b = 12 + (-16) = -4
\]

Thus, the value of \(a + b\) is \(\boxed{-4}\).

Would you like more details on any step, or further clarifications?

Here are 5 related questions for further practice:
1. What happens if the two curves have different slopes at the same point?
2. How do you find the point of tangency for a quadratic curve?
3. What is the general condition for a line to be tangent to a curve at a given point?
4. Can you find the value of \(t\) if the curves are \(y = x^3\) and \(y = 5x^2 - 2\)?
5. How would the answer change if the curve was \(y = x^4\) instead of \(y = x^3\)?

**Tip**: Always verify the slopes when solving tangency problems involving multiple curves to ensure the tangency condition is met.

두 곡선 y = x^3, y = 3x^2 - 4가 x = t에서 접선 y = ax + b를 공통으로 가질 때, 상수 a, b의 합 a + b의 값은?

Learn how to find the sum of constants a + b for a tangent line y = ax + b that is common to two curves y = x^3 and y = 3x^2 - 4 at x = t. This problem covers key calculus concepts, including tangency conditions, derivatives, and solving a system of equations. Suitable for students in Grades 10-12.

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