We already know the roots are x= -4 and x=2.5 so use that too

Given that the quadratic function \( y = 2x^2 + 3x - 20 \) has roots at \( x = -4 \) and \( x = \frac{5}{2} \), we can use these roots to find the vertex.

The vertex of a parabola is located exactly halfway between its roots. To find the x-coordinate of the vertex, take the average of the roots:

\[ x_{\text{vertex}} = \frac{-4 + \frac{5}{2}}{2} \]

First, convert \(-4\) to a fraction with the same denominator as \(\frac{5}{2}\):

\[ -4 = \frac{-8}{2} \]

Now, add the fractions:

\[ x_{\text{vertex}} = \frac{\frac{-8}{2} + \frac{5}{2}}{2} = \frac{\frac{-8 + 5}{2}}{2} = \frac{\frac{-3}{2}}{2} = \frac{-3}{4} \]

So the x-coordinate of the vertex is \( x = -\frac{3}{4} \).

Next, substitute \( x = -\frac{3}{4} \) back into the original quadratic equation to find the y-coordinate of the vertex:

\[ y = 2\left(-\frac{3}{4}\right)^2 + 3\left(-\frac{3}{4}\right) - 20 \]

Calculate each term:
\[ 2\left(-\frac{3}{4}\right)^2 = 2 \cdot \frac{9}{16} = \frac{18}{16} = \frac{9}{8} \]
\[ 3\left(-\frac{3}{4}\right) = -\frac{9}{4} \]

Combine these results:
\[ y = \frac{9}{8} - \frac{9}{4} - 20 \]

Convert \(\frac{9}{4}\) to \(\frac{18}{8}\) to combine:
\[ y = \frac{9}{8} - \frac{18}{8} - 20 = \frac{9 - 18}{8} - 20 = -\frac{9}{8} - 20 \]

Convert \(-20\) to eighths:
\[ -20 = -\frac{160}{8} \]

Combine:
\[ y = -\frac{9}{8} - \frac{160}{8} = -\frac{169}{8} \]

So, the vertex of the quadratic function is:
\[ \left( -\frac{3}{4}, -\frac{169}{8} \right) \]

Would you like more details or have any questions about this problem? Here are some questions you might consider:

1. How do you find the vertex of a quadratic function by completing the square?
2. What are some applications of finding the vertex of a quadratic function?
3. How can you determine if a vertex is a maximum or minimum?
4. How do you derive the quadratic formula from completing the square?
5. Can you explain the significance of the axis of symmetry in a parabola?

**Tip:** Knowing the roots of a quadratic function simplifies finding the vertex, as the vertex lies exactly halfway between the roots.

Learn how to find the vertex of the quadratic function y = 2x^2 + 3x - 20 using given roots. This problem involves using the vertex formula and completing the square method. Suitable for students in Grades 9-12.

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