Quadratic Equation

Quadratic equations looks like: ax² + bx + c = 0

where a,b,c are real numbers, and a ≠ 0. Every quadratic equation can have 0, 1 or 2 real decidions derived by the formula:

The number D = b² - 4ac is called discriminant.

If D < 0 then the quadratic equation have no decidions. If D = 0 then the quadratic equation have 1 decidion x = - ^b/_2a. If D > 0 then the quadratic equation have 2 decidions.

Example:

If we have equation: x² + 3x - 4 = 0

a = 1, b = 3, c = -4

Parabola

The graph of a quadratic equatin is called a parabola.

If a > 0 then graph horns pointing down:

if a < 0 then graph horns pointing up:

The midpoint of any parabola is the point x = -b/2a.

Sign of Quadratic Equation

Let f(x) = ax^2 + bx + c , where a,b,c &# 949; R and a ≠ 0

Vieta's formulas

If x₁ and x₂ are the roots of the quadratic equation ax² + bx + c = 0 then:

These formulas are called Vieta's formulas.

We can find the roots x₁ and x₂ of a quadratic equation by solving the system above.

Maxima and minima

Maxima a>0  x= - b/2a  Max = -(b^2 -4ac)/4a

Minima a<0  -----------------------------------------(same)