When the coefficients a, b, and c, are real or complex, the roots are

[image]

where the discriminant is defined as

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In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.

f(x)=ax²+bx+c

with a, b, and c are real #'s

If a>0 then parabala opens upward, concave up

If a<0 then parabala opens downward, concave down

x=-b/2a

(-b/2a, f(-b/2a))

A. When the graph opens down the vertex is a max 

B. When the graph opens up the vertex is a min

Increasing function (-∞, x]

Decreasing function [x, -∞)

Vertex form of a parabola with vertex (h,k)

Y=a(x-h)² + k

1. Add and subtract (1/2b)² from the right hand side of the equation

2. Factor the perfect square trinomial

1. The Quadratic Function of a parabola is y=a(x-h)²+k
2. Substitute the coordinates of the vertex (turning point) into the formula

3. Substitute any point (x,y) where the parabola intersects the x axis for a

4. Solve the equations for a

5. Substitute answer for a into the formula to get the Quadratic Function of the parabola