To solve the quadratic equation \( x^2 + 10x - 39 = 0 \), we can use the **quadratic formula**, which is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation, \( a = 1 \), \( b = 10 \), and \( c = -39 \).

Let's plug these values into the quadratic formula:

\[
x = \frac{-10 \pm \sqrt{10^2 - 4(1)(-39)}}{2(1)}
\]

Simplifying:

\[
x = \frac{-10 \pm \sqrt{100 + 156}}{2}
\]
\[
x = \frac{-10 \pm \sqrt{256}}{2}
\]
\[
x = \frac{-10 \pm 16}{2}
\]

Now, calculate the two possible solutions:

1. \( x = \frac{-10 + 16}{2} = \frac{6}{2} = 3 \)
2. \( x = \frac{-10 - 16}{2} = \frac{-26}{2} = -13 \)

So, the two solutions are:

\[
x = 3 \quad \text{or} \quad x = -13
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What happens when a quadratic equation has no real roots?
2. Can you factor this quadratic equation instead of using the quadratic formula?
3. How do you know if a quadratic equation will have two, one, or no real solutions?
4. How does completing the square compare to using the quadratic formula?
5. Can you graph this quadratic function and identify the roots visually?

**Tip:** The discriminant \( b^2 - 4ac \) helps you determine the nature of the roots: if it's positive, there are two real roots; if zero, one real root; and if negative, no real roots.

Discover how to solve the quadratic equation x² + 10x - 39 step-by-step using the quadratic formula. This detailed explanation covers core algebraic concepts and the quadratic formula theorem. Ideal for students in Grades 8-10.

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