The Quadratic Formula

ax²+ bx + c = 0

The quadratic formula is used to solve unfactorable quadratic equations.

ex.
x² + 3x + 6 = 0
a =1; b = 3; c = 6.

To check to see if the roots are correct for the equation, use the sum and product rule of two roots. The sum of the roots produces the relationship of -b/a.

ex.
x² + 3x + 6 = 0

a = 1, b = 3, c = 6

-b/a = -3/1 = -3
( The sum of the roots)

The product of the roots produces the relationship c/a.

b² - 4ac is called the discriminant. It provides a way to find roots for the quadratic formula without solving the whole equation.

If b² - 4ac = 0 then, the equation has a double root or one real solution.
If b² - 4ac > 0 then, the equation has two real but unequal roots.
If b² - 4ac < 0 then, the equation has two nonreal, complex, unequal roots; which are complex conjugates of each other.

ex.
2x² - 4x + 2 = 0
a = 2, b = -4, c = 2

b² - 4ac
(-4)² - 4(2)(2)
16 - 16 = 0
b² - 4ac = 0

there are two real and equal roots.

3x² - 5x - 2 = 0
a = 3, b = -5, c = -2

b² - 4ac
(-5)2 - 4(3)(-2)
25 - (-24) = 25 + 24 = 49
b² - 4ac > 0

there are two real and unequal roots.

5x² + 4x + 1 = 0
a = 5, b = 4, c = 1

b² - 4ac
(4)2 - 4(5)(1)
16 - 20 = -4
b² - 4ac < 0

there are two nonreal and complex roots.