Find all the values of a where the tangent line to y=x^3 at x=a is parallel to the tangent line to y=x^2+5x at x=a

We are tasked with finding all values of \( a \) such that the tangent lines to the curves \( y = x^3 \) and \( y = x^2 + 5x \) at \( x = a \) are parallel. This means the slopes of these tangent lines must be equal at \( x = a \).

### Step 1: Find the derivatives of the two functions.
The derivative of a function represents the slope of the tangent line at any point.

1. For \( y = x^3 \), the derivative is:
   \[
   \frac{dy}{dx} = 3x^2
   \]
   Therefore, the slope of the tangent line to \( y = x^3 \) at \( x = a \) is:
   \[
   \text{slope}_1 = 3a^2
   \]

2. For \( y = x^2 + 5x \), the derivative is:
   \[
   \frac{dy}{dx} = 2x + 5
   \]
   Therefore, the slope of the tangent line to \( y = x^2 + 5x \) at \( x = a \) is:
   \[
   \text{slope}_2 = 2a + 5
   \]

### Step 2: Set the slopes equal.
Since we want the tangent lines to be parallel, the slopes must be equal at \( x = a \). Therefore, set the two slopes equal:
\[
3a^2 = 2a + 5
\]

### Step 3: Solve for \( a \).
Rearrange the equation:
\[
3a^2 - 2a - 5 = 0
\]
This is a quadratic equation. We can solve it using the quadratic formula:
\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 3 \), \( b = -2 \), and \( c = -5 \).

Plugging in these values:
\[
a = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-5)}}{2(3)}
\]
\[
a = \frac{2 \pm \sqrt{4 + 60}}{6}
\]
\[
a = \frac{2 \pm \sqrt{64}}{6}
\]
\[
a = \frac{2 \pm 8}{6}
\]
Thus, the two possible values of \( a \) are:
\[
a = \frac{2 + 8}{6} = \frac{10}{6} = \frac{5}{3}
\]
\[
a = \frac{2 - 8}{6} = \frac{-6}{6} = -1
\]

### Conclusion:
The values of \( a \) where the tangent lines are parallel are:
\[
a = \frac{5}{3} \quad \text{and} \quad a = -1
\]

Would you like more details on any part of the process? 

Here are some related questions:
1. What are the geometric interpretations of parallel tangent lines?
2. How does the quadratic formula work in this context?
3. What would happen if the equations were changed to higher-degree polynomials?
4. Can we generalize this approach to other types of curves?
5. How does the choice of \( a \) affect the actual tangent line equations?

**Tip:** In problems involving tangent lines, derivatives are your best tool for analyzing the slopes at specific points!

Explore how to find values of a where the tangent lines to y=x^3 and y=x^2+5x are parallel. This problem involves key calculus concepts like derivatives and solving quadratic equations. Suitable for students in Grades 11-12.

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