solve x ^ 2 + 10 ^ -20x - 10 ^ - 20 = 0 by middle term splitting tell correct answer 

Apologies for the confusion earlier. The approach of using middle term splitting requires finding two terms that multiply to the constant term and add up to the middle coefficient. Let’s carefully solve the given equation using middle term splitting.

The equation given is:

\[
x^2 + 10^{-20}x - 10^{-20} = 0.
\]

To split the middle term correctly, let's rewrite it with better notation:

\[
x^2 + (10^{-20})x - 10^{-20} = 0.
\]

### Step 1: Identify factors

We need two numbers that:

1. Multiply to \(-10^{-20}\) (the product of the constant term).
2. Add up to \(10^{-20}\) (the middle coefficient).

### Step 2: Factoring by splitting the middle term

In this case, let's attempt to split \(10^{-20}\) into suitable factors:

The factors \(10^{-20}\) and \(0\) would work since:

- Their product is \(0 \times 10^{-20} = 0\).
- Their sum is \(10^{-20} + 0 = 10^{-20}\).

However, since we see this isn’t directly feasible using classical factors due to their extreme size, it's evident that quadratic methods are used here more effectively:

Splitting into:

\[
(x - 10^{-10})(x + 10^{-10}) = 0,
\]

which gives the correct solutions. 

So the factored and precise answer without guessing steps would be as follows:

\[
x = 10^{-10} \text{ and } x = -10^{-10}.
\]

Is that clear? Let me know any clarifications and further refinements you want!

### Questions to Explore
1. What happens if coefficients are nearly zero?
2. Is splitting always possible with tiny decimals?
3. Methods: Completing squares or quadratic formula?
4. When does exact factorization get difficult?
5. Why do substitutions make middle splitting easier?

**Tip:** With unusual coefficients, solving and simplification alternatives often lead quicker!

Learn how to solve the quadratic equation x^2 + 10^-20x - 10^-20 = 0 using the middle term splitting method. This problem covers advanced algebraic concepts involving quadratic equations with very small coefficients. Suitable for students in Grades 10-12.

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