The equation you provided is a quadratic equation in the form:

\[
y = x^2 + 7x - 30
\]

We can analyze this equation in several ways, including finding its roots, vertex, and the direction of its parabola. Here's a breakdown:

### 1. **Finding the Roots (x-intercepts)**:
To find the values of \(x\) where \(y = 0\), we solve the quadratic equation:

\[
x^2 + 7x - 30 = 0
\]

We can solve this using the quadratic formula:

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

For the equation \(x^2 + 7x - 30 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 7\)
- \(c = -30\)

Substituting into the formula:

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(1)(-30)}}{2(1)}
\]
\[
x = \frac{-7 \pm \sqrt{49 + 120}}{2}
\]
\[
x = \frac{-7 \pm \sqrt{169}}{2}
\]
\[
x = \frac{-7 \pm 13}{2}
\]

So the two possible solutions are:
\[
x = \frac{-7 + 13}{2} = 3 \quad \text{and} \quad x = \frac{-7 - 13}{2} = -10
\]

The roots are:
\[
x = 3 \quad \text{and} \quad x = -10
\]

### 2. **Finding the Vertex**:
The vertex of a parabola given by \(y = ax^2 + bx + c\) is located at \(x = -\frac{b}{2a}\). Substituting \(a = 1\) and \(b = 7\):

\[
x = -\frac{7}{2(1)} = -\frac{7}{2} = -3.5
\]

To find the \(y\)-coordinate of the vertex, substitute \(x = -3.5\) into the original equation:

\[
y = (-3.5)^2 + 7(-3.5) - 30
\]
\[
y = 12.25 - 24.5 - 30 = -42.25
\]

The vertex is \((-3.5, -42.25)\).

### 3. **Direction of the Parabola**:
Since the coefficient of \(x^2\) is positive (\(a = 1\)), the parabola opens upwards.

---

Would you like more details or have any questions?

### Relative Questions:
1. How would you graph the equation \(y = x^2 + 7x - 30\)?
2. What happens to the graph if you change the constant term in the equation?
3. How do you complete the square for this quadratic equation?
4. What are the applications of quadratic equations in real life?
5. How does the discriminant help determine the number of real roots?

### Tip:
Always check the sign of the leading coefficient (\(a\)) in a quadratic equation to quickly determine the direction the parabola opens (up if \(a > 0\), down if \(a < 0\)).

Explore how to solve the quadratic equation y = x^2 + 7x - 30 with detailed solutions, including finding the roots using the quadratic formula, determining the vertex, and analyzing the direction of the parabola. Suitable for students in Grades 8-10.

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