(x%2B4)-in-vertex-form.jpeg "(x+5)(x+4) In Vertex Form")
From Factored Form to Vertex Form: (x + 5)(x + 4)
This article will guide you through the process of converting the quadratic expression (x + 5)(x + 4) from factored form to vertex form.
Understanding Vertex Form
The vertex form of a quadratic equation is given by:
f(x) = a(x - h)² + k
Where:
- a determines the direction and width of the parabola
- (h, k) represents the coordinates of the vertex
Steps to Convert
-
Expand the factored form:
(x + 5)(x + 4) = x² + 9x + 20
-
Complete the square:
- Take half of the coefficient of the x term (9/2), square it (81/4), and add and subtract it inside the expression:
x² + 9x + 20 = x² + 9x + 81/4 - 81/4 + 20
- Rewrite the first three terms as a perfect square trinomial:
(x + 9/2)² - 81/4 + 20
-
Simplify:
(x + 9/2)² - 1/4
-
Vertex Form:
The expression is now in vertex form:
f(x) = (x + 9/2)² - 1/4
Identifying the Vertex
By comparing this equation to the general vertex form, we can see that:
- a = 1
- h = -9/2
- k = -1/4
Therefore, the vertex of the parabola represented by the equation (x + 5)(x + 4) is (-9/2, -1/4).
Key Takeaways
- Converting from factored form to vertex form allows you to easily identify the vertex of the parabola.
- Completing the square is a crucial step in this process.
- Vertex form provides a clear understanding of the parabola's shape, orientation, and position on the coordinate plane.