To solve the quadratic equation \( a^2 + 2a = 203119913336830 \), we can use the quadratic formula. The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

In this case:

\[
a^2 + 2a - 203119913336830 = 0
\]

So, we have:

- \( A = 1 \)
- \( B = 2 \)
- \( C = -203119913336830 \)

The quadratic formula is given by:

\[
a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\]

Substituting the values:

\[
a = \frac{-2 \pm \sqrt{2^2 - 4(1)(-203119913336830)}}{2(1)}
\]

Now, let's calculate this.The solutions to the quadratic equation \( a^2 + 2a = 203119913336830 \) are:

\[
a_1 \approx 14,252,013.36
\]
\[
a_2 \approx -14,252,015.36
\]

Since \(a\) is often taken as a positive value in practical scenarios, the primary solution is:

\[
a \approx 14,252,013.36
\]

Would you like more details or have any questions?

Here are 5 related questions:

1. How can we verify the solutions by substituting back into the original equation?
2. What is the significance of the discriminant in a quadratic equation?
3. How does the quadratic formula derive from completing the square?
4. Are there any specific applications of quadratic equations in real-world problems?
5. How would the solutions change if the coefficient of \(a^2\) were different?

**Tip:** When dealing with large numbers in quadratic equations, make sure your calculator or software can handle large values accurately to avoid rounding errors.

Explore the step-by-step solution to the quadratic equation a^2 + 2a = 203119913336830. This problem demonstrates the application of the quadratic formula and is suitable for students in Grades 10-12.

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