The equation $x^2 + 6x - 27 = 0$ is a quadratic equation. To solve it, we'll use the quadratic formula:

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

For the equation $x^2 + 6x - 27 = 0$, the coefficients are:

• $a = 1$
• $b = 6$
• $c = -27$

Substitute these values into the quadratic formula:

$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-27)}}{2(1)}$

Simplifying this:

$x = \frac{-6 \pm \sqrt{36 + 108}}{2}$

$x = \frac{-6 \pm \sqrt{144}}{2}$

$x = \frac{-6 \pm 12}{2}$

This gives us two possible solutions:

$x_1 = \frac{-6 + 12}{2} = \frac{6}{2} = 3$

$x_2 = \frac{-6 - 12}{2} = \frac{-18}{2} = -9$

So, the solutions to the equation $x^2 + 6x - 27 = 0$ are:

$x = 3 \quad \text{or} \quad x = -9$

Would you like further details or have any questions?

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Here are some related questions:

1. How can you verify the solutions of a quadratic equation?
2. What are the steps to solve a quadratic equation by factoring?
3. How does the discriminant determine the nature of the roots?
4. What is the axis of symmetry for a quadratic function?
5. How can you use the quadratic formula to solve equations with complex roots?
6. What is the significance of the vertex in the graph of a quadratic function?
7. How do you solve a quadratic equation by completing the square?
8. What are some practical applications of solving quadratic equations?

Tip: Always double-check your calculations, especially when working with square roots, to avoid errors in finding the roots of a quadratic equation.