find vertex for the equation for y=(x+6)(2x-5)

To find the vertex of the quadratic equation \( y = (x + 6)(2x - 5) \), follow these steps:

1. **Expand the equation**:
   \[
   y = (x + 6)(2x - 5)
   \]
   Use the distributive property (FOIL):
   \[
   y = x(2x - 5) + 6(2x - 5)
   \]
   \[
   y = 2x^2 - 5x + 12x - 30
   \]
   \[
   y = 2x^2 + 7x - 30
   \]

   Now the equation is in the standard quadratic form \( y = ax^2 + bx + c \), where \( a = 2 \), \( b = 7 \), and \( c = -30 \).

2. **Find the x-coordinate of the vertex**:
   The x-coordinate of the vertex is given by the formula:
   \[
   x = \frac{-b}{2a}
   \]
   Substituting \( a = 2 \) and \( b = 7 \):
   \[
   x = \frac{-7}{2(2)} = \frac{-7}{4}
   \]

3. **Find the y-coordinate of the vertex**:
   Substitute \( x = \frac{-7}{4} \) into the equation \( y = 2x^2 + 7x - 30 \) to find the y-coordinate:
   \[
   y = 2\left(\frac{-7}{4}\right)^2 + 7\left(\frac{-7}{4}\right) - 30
   \]
   Simplifying:
   \[
   y = 2\left(\frac{49}{16}\right) + 7\left(\frac{-7}{4}\right) - 30
   \]
   \[
   y = \frac{98}{16} + \frac{-49}{4} - 30
   \]
   \[
   y = \frac{98}{16} - \frac{196}{16} - 30
   \]
   \[
   y = \frac{-98}{16} - 30 = \frac{-49}{8} - 30
   \]
   Converting 30 into a fraction:
   \[
   y = \frac{-49}{8} - \frac{240}{8}
   \]
   \[
   y = \frac{-289}{8}
   \]

Thus, the vertex of the parabola is \( \left( \frac{-7}{4}, \frac{-289}{8} \right) \).

Would you like further details on any part of this?

Here are 5 related questions to explore:
1. How do you convert a quadratic equation into vertex form?
2. What does the vertex represent on the graph of a parabola?
3. How do you find the axis of symmetry for a quadratic equation?
4. What is the significance of the discriminant in a quadratic equation?
5. How do you complete the square to find the vertex?

**Tip:** The vertex of a parabola is either its highest or lowest point, depending on whether it opens upward or downward.

Learn how to find the vertex of the quadratic equation y=(x+6)(2x-5). This step-by-step guide covers key algebraic concepts, including expanding the equation, using the vertex formula, and solving for both x and y coordinates. Ideal for students in Grades 9-10.

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