The formula [tex]\( x = \frac{b}{2a} \)[/tex] is used to find the [tex]\( x \)[/tex]-value of the vertex of a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex].
To understand this in detail, let's break it down step by step:
1. Quadratic Equation Form: A quadratic equation is typically written in the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable.
2. Vertex of a Parabola: The graph of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is a parabola. The vertex of this parabola is the highest or lowest point on the graph, depending on the direction in which it opens. If [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the vertex is the minimum point. If [tex]\( a < 0 \)[/tex], the parabola opens downwards, and the vertex is the maximum point.
3. Vertex Formula: The [tex]\( x \)[/tex]-value of the vertex of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = \frac{b}{2a} \)[/tex]. This is derived from the general form of a quadratic function and the process of completing the square or using calculus to find critical points.
4. Step-by-Step Application:
- Identify the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the quadratic equation.
- Plug these coefficients into the formula [tex]\( x = \frac{b}{2a} \)[/tex].
- Simplify the expression to find the [tex]\( x \)[/tex]-coordinate of the vertex.
By following these basic steps, you can determine the [tex]\( x \)[/tex]-value of the vertex of any quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex].
