To solve the quadratic equation \(6x^2 - 27x + 30 = 0\), we can use the quadratic formula:

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

where \(a = 6\), \(b = -27\), and \(c = 30\).

1. **Calculate the discriminant** \(D = b^2 - 4ac\):
   \[
   D = (-27)^2 - 4 \cdot 6 \cdot 30 = 729 - 720 = 9
   \]

2. **Use the quadratic formula**:
   \[
   x = \frac{-(-27) \pm \sqrt{9}}{2 \cdot 6} = \frac{27 \pm 3}{12}
   \]

3. **Find the two solutions**:
   - \(x_1 = \frac{27 + 3}{12} = \frac{30}{12} = \frac{5}{2} = 2.5\)
   - \(x_2 = \frac{27 - 3}{12} = \frac{24}{12} = 2\)

Thus, the solutions are \(x = 2.5\) and \(x = 2\).

Would you like further details on any specific step, or do you have any questions? Here are some related questions you might consider:

1. How do you factor the quadratic equation \(6x^2 - 27x + 30\)?
2. What is the vertex of the parabola represented by this equation?
3. How can we graph the equation \(6x^2 - 27x + 30\)?
4. What are the implications of the discriminant being positive?
5. How would the solutions change if the equation had different coefficients?

**Tip**: Always check the solutions by substituting them back into the original equation to verify their accuracy.

Explore the step-by-step solution for the quadratic equation 6x² - 27x + 30, using the quadratic formula. This comprehensive guide includes the calculation of the discriminant and finding both solutions, suitable for Grades 7-9 students.

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