91Ó°ÊÓ

Quadratic equations are polynomial equations of degree two. They follow the general form of \( ax^2 + bx + c = 0 \).
Here, \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

These equations graph as parabolas, which can open either upward or downward.
The shape of the parabola and the direction in which it opens depend on the coefficient \(a\).

If \(a > 0\), the parabola opens upward. If \(a < 0\), it opens downward.
Quadratic equations are fundamental in various areas of math and science, especially in physics for describing motions and in engineering for analyzing parabolic structures.

The vertex formula is used to find the vertex of a parabola when the quadratic equation is in the form \( y = ax^2 + bx + c \).
The vertex of a parabola is its highest or lowest point, known as the maximum or minimum.
For a parabola opening upward (when \(a > 0\)) or downward (when \(a < 0\)), its vertex can be found using the x-coordinate formula:

• \( x = -\frac{b}{2a} \)

Once we have the x-coordinate, substitute this value back into the original equation to find the y-coordinate of the vertex:

• \( y = g(x) \)

In the given example, we identified that \(a = 4\), \(b = -64\), and \(c = 107\). By inserting these into the vertex formula, we calculated the vertex' coordinates, \( (8, -149) \).

Coordinate geometry, also known as analytic geometry, allows us to study geometry using a coordinate system.
In this system, every point has coordinates \((x, y)\) that describe its location on a plane.

For quadratic equations, coordinate geometry lets us graph the equation as a parabola and analyze its shape, position, and dimensions.
We can find key features like vertices, intercepts, and axes of symmetry.

For example, a parabola's vertex gives us important information about its peak or trough, while the x- and y-intercepts show where it crosses the coordinate axes.
This application is powerful in solving various real-world problems, from engineering to economics.

Parabolas are U-shaped graphs of quadratic equations. Each parabola is symmetric about a vertical line called the axis of symmetry.
The vertex of a parabola is its highest or lowest point and represents the extremum of the quadratic function.

Parabolas can model various phenomena in physics and engineering.
For example, the path of a projectile under gravity follows a parabolic trajectory, and satellite dishes are shaped like parabolas to focus signals.
Understanding the features of parabolas helps in these practical applications.
The direction in which a parabola opens (upward or downward) is determined by the sign of the quadratic term's coefficient \(a\).
For the given equation, we determined that the parabola opens upward and the vertex is at \( (8, -149) \).
This comprehensive understanding of parabolas enriches our ability to analyze and solve related mathematical problems.