The equation $x^2 + 10x - 5 = 0$ cannot be factored easily using simple factorization techniques due to the non-perfect nature of the constants. However, we can solve it using the quadratic formula:

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

For the equation $x^2 + 10x - 5 = 0$, the coefficients are:

• $a = 1$
• $b = 10$
• $c = -5$

Now, we substitute these values into the quadratic formula:

$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(-5)}}{2(1)}$ $x = \frac{-10 \pm \sqrt{100 + 20}}{2}$ $x = \frac{-10 \pm \sqrt{120}}{2}$ $x = \frac{-10 \pm 2\sqrt{30}}{2}$ $x = -5 \pm \sqrt{30}$

Thus, the solutions are:

$x = -5 + \sqrt{30} \quad \text{or} \quad x = -5 - \sqrt{30}$

Would you like a detailed explanation of how the quadratic formula works or more insights on solving similar equations?

Here are 5 related questions you might want to explore:

1. How do you complete the square for quadratic equations?
2. What is the discriminant in the quadratic formula and what does it tell us?
3. Can you explain how to factor quadratic expressions with perfect squares?
4. How can you derive the quadratic formula?
5. What is the significance of the roots of a quadratic equation?

Tip: When the quadratic equation does not factor nicely, using the quadratic formula is a foolproof method to solve it!